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            Lecture 22: Expectation I
         
    
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<p><strong>Description:</strong> Covers expected value as it relates to random variables, discussing coin games, network latency, and the hat check problem.</p> <p><strong>Speaker:</strong> Tom Leighton</p>
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<p><span m="360">The</span> <span m="490">following</span> <span m="940">content</span> <span m="1530">is</span> <span m="1650">provided</span> <span m="2100">under</span> <span m="2350">a</span> <span m="2400">Creative</span> <span m="2800">Commons</span> <span m="3220">license.</span> <span m="4340">Your</span> <span m="4510">support</span> <span m="5050">will</span> <span m="5200">help</span> <span m="5430">MIT</span> <span m="5870">OpenCourseWare</span> <span m="6660">continue</span> <span m="7170">to</span> <span m="7290">offer</span> <span m="7690">high</span> <span m="7910">quality</span> <span m="8420">educational</span> <span m="9060">resources</span> <span m="9640">for</span> <span m="9820">free.</span> <span m="11020">To</span> <span m="11040">make</span> <span m="11250">a</span> <span m="11290">donation</span> <span m="11980">or</span> <span m="12230">view</span> <span m="12680">additional</span> <span m="13120">materials</span> <span m="13640">from</span> <span m="13830">hundreds</span> <span m="14240">of</span> <span m="14360">MIT</span> <span m="14760">courses,</span> <span m="15840">visit</span> <span m="16100">MIT</span> <span m="16520">OpenCourseWare</span> <span m="17560">at</span> <span m="17750">ocw.mit.edu.</span> </p>
<p><span m="24780">PROFESSOR: OK,</span> <span m="25700">let's</span> <span m="25860">get</span> <span m="26030">started.</span> <span m="29400">Last</span> <span m="29750">week</span> <span m="30020">we</span> <span m="30190">talked</span> <span m="30550">about</span> <span m="30950">random</span> <span m="31350">variables.</span> <span m="32520">And</span> <span m="33210">this</span> <span m="33520">week</span> <span m="33920">we're</span> <span m="34020">going</span> <span m="34140">to</span> <span m="34220">talk</span> <span m="34540">about</span> <span m="35100">their</span> <span m="35360">expected</span> <span m="36180">value.</span> <span m="37550">The</span> <span m="37900">expected</span> <span m="38440">value</span> <span m="38770">of</span> <span m="38840">a</span> <span m="38940">random</span> <span m="39340">variable</span> <span m="41000">comes</span> <span m="41290">up</span> <span m="41420">in</span> <span m="41500">all</span> <span m="41790">sorts</span> <span m="42050">of</span> <span m="42150">applications.</span> <span m="42830">We're</span> <span m="42890">going</span> <span m="43010">to</span> <span m="43070">spend</span> <span m="43380">a</span> <span m="43430">whole</span> <span m="43660">week</span> <span m="43890">talking</span> <span m="44240">about</span> <span m="44680">it and</span> <span m="44910">some of</span> <span m="45130">next</span> <span m="45410">week</span> <span m="45580">talking</span> <span m="45860">about</span> <span m="46100">variations</span> <span m="46980">from</span> <span m="47250">the</span> <span m="47330">expected</span> <span m="47770">value.</span> <span m="48580">And</span> <span m="48770">it's</span> <span m="48890">probably</span> <span m="49210">one</span> <span m="49350">of</span> <span m="49410">the</span> <span m="49480">best</span> <span m="49870">tools</span> <span m="50290">you</span> <span m="50410">have</span> <span m="50870">for</span> <span m="50980">working</span> <span m="51370">with</span> <span m="51640">probability</span> <span m="52190">problems</span> <span m="52750">in</span> <span m="52840">practice.</span> </p>
<p><span m="54460">So</span> <span m="54570">let's</span> <span m="54800">write</span> <span m="54980">down</span> <span m="55190">the</span> <span m="55260">definition.</span> <span m="64129">The</span> <span m="64379">expected</span> <span m="65650">value</span> <span m="70200">also</span> <span m="70490">has</span> <span m="70620">other</span> <span m="70780">names.</span> <span m="72320">It's</span> <span m="72510">also</span> <span m="72890">known</span> <span m="73200">as</span> <span m="73810">the</span> <span m="74500">average</span> <span m="78390">or</span> <span m="78800">the</span> <span m="78920">mean</span> <span m="83570">of</span> <span m="83740">a</span> <span m="83840">random</span> <span m="84230">variable.</span> <span m="89390">Random</span> <span m="89700">variable</span> <span m="90100">R</span> <span m="91510">over</span> <span m="92380">a</span> <span m="92540">probability</span> <span m="93300">space</span> <span m="93790">S</span> <span m="101950">is</span> <span m="102450">denoted</span> <span m="102960">by</span> <span m="103680">a</span> <span m="103830">lot</span> <span m="104100">of</span> <span m="104180">ways.</span> <span m="104520">We're</span> <span m="104620">going</span> <span m="104750">to</span> <span m="104820">use</span> <span m="105120">Ex</span> <span m="105670">to</span> <span m="105770">denote</span> <span m="106140">it,</span> <span m="106720">Ex</span> <span m="107150">of</span> <span m="107270">R.</span> </p>
<p><span m="108760">And</span> <span m="108970">it's</span> <span m="109130">the</span> <span m="109230">sum</span> <span m="109870">over</span> <span m="110140">all</span> <span m="110310">possible</span> <span m="110930">outcomes</span> <span m="111510">in</span> <span m="111590">the</span> <span m="111650">sample</span> <span m="112070">space</span> <span m="113340">of</span> <span m="113540">the</span> <span m="113610">value</span> <span m="114700">of</span> <span m="114900">the</span> <span m="115000">random</span> <span m="115310">variable</span> <span m="115670">on</span> <span m="115810">that</span> <span m="115970">outcome</span> <span m="116950">times</span> <span m="117430">the</span> <span m="117510">probability</span> <span m="118810">of</span> <span m="118980">that</span> <span m="119170">outcome.</span> <span m="121190">In</span> <span m="121240">other</span> <span m="121410">words,</span> <span m="121780">the</span> <span m="121880">expected</span> <span m="122350">value</span> <span m="123390">of</span> <span m="123520">a</span> <span m="123610">random</span> <span m="123900">variable</span> <span m="124310">is</span> <span m="124380">just</span> <span m="124570">a</span> <span m="124630">weighted</span> <span m="125160">average</span> <span m="126590">of</span> <span m="126800">all</span> <span m="127160">possible</span> <span m="127770">values</span> <span m="129080">of</span> <span m="129259">the</span> <span m="129360">random</span> <span m="129690">variable</span> <span m="130729">where</span> <span m="130880">the</span> <span m="130970">weight</span> <span m="131580">is</span> <span m="131780">the</span> <span m="131870">probability</span> <span m="133380">of</span> <span m="133590">that</span> <span m="133800">happening.</span> </p>
<p><span m="135910">For</span> <span m="136070">example,</span> <span m="139950">suppose</span> <span m="140500">we</span> <span m="140640">roll</span> <span m="143410">a</span> <span m="143590">fair</span> <span m="145980">six</span> <span m="146560">sided</span> <span m="147000">die.</span> <span m="150860">So</span> <span m="151010">the</span> <span m="151100">numbers</span> <span m="151460">that</span> <span m="151550">come</span> <span m="151690">up</span> <span m="151850">are</span> <span m="151910">1</span> <span m="152190">to</span> <span m="152260">6.</span> <span m="153810">And</span> <span m="154060">we</span> <span m="154170">let</span> <span m="154500">R</span> <span m="155930">be</span> <span m="156260">the</span> <span m="156660">random</span> <span m="157040">variable</span> <span m="157440">denoting</span> <span m="157890">the</span> <span m="158000">outcome,</span> <span m="159150">1</span> <span m="159450">through</span> <span m="159640">6.</span> <span m="163330">So</span> <span m="163510">the</span> <span m="163660">expected</span> <span m="164250">value</span> <span m="164780">of</span> <span m="164870">the</span> <span m="164970">role,</span> <span m="165990">we</span> <span m="166140">can</span> <span m="166420">easily</span> <span m="166780">compute</span> <span m="167180">from</span> <span m="167300">the</span> <span m="167360">definition.</span> <span m="170160">Well,</span> <span m="170400">there's</span> <span m="170620">a</span> <span m="170670">1/6</span> <span m="171530">chance</span> <span m="172200">that</span> <span m="172300">it</span> <span m="172360">comes</span> <span m="172640">out a</span> <span m="172780">1.</span> <span m="174890">The</span> <span m="174970">next</span> <span m="175240">outcome</span> <span m="175540">would</span> <span m="175610">be</span> <span m="175760">a</span> <span m="175810">2.</span> <span m="176200">That</span> <span m="176390">happens</span> <span m="176760">with</span> <span m="176890">1/6</span> <span m="177360">chance.</span> <span m="179600">3,</span> <span m="179880">4,</span> <span m="180090">5,</span> <span m="180410">and</span> <span m="180530">six</span> <span m="181800">all</span> <span m="182090">happen</span> <span m="182390">with</span> <span m="182500">a 1/6</span> <span m="182570">chance.</span> <span m="186680">If</span> <span m="186770">I</span> <span m="186870">sum</span> <span m="187150">up</span> <span m="187300">1,</span> <span m="187520">2,</span> <span m="187680">3,</span> <span m="187990">4</span> <span m="188280">up</span> <span m="188400">to</span> <span m="188480">6,</span> <span m="188970">I</span> <span m="189080">get</span> <span m="189870">6</span> <span m="190180">times</span> <span m="190450">7</span> <span m="190790">over</span> <span m="191010">2</span> <span m="192660">times</span> <span m="193000">the</span> <span m="193090">1/6</span> <span m="194400">is</span> <span m="194670">just</span> <span m="195050">7</span> <span m="195340">over</span> <span m="195540">2</span> <span m="195870">or</span> <span m="196000">3</span> <span m="196300">and</span> <span m="196600">1/2.</span> </p>
<p><span m="199590">So</span> <span m="199660">expected</span> <span m="200230">value</span> <span m="200640">when</span> <span m="200740">I</span> <span m="200800">roll</span> <span m="201120">a</span> <span m="201180">die,</span> <span m="202050">a</span> <span m="202140">fair</span> <span m="202440">six</span> <span m="202660">sided</span> <span m="202950">die,</span> <span m="203320">is</span> <span m="203450">just</span> <span m="203650">3</span> <span m="204200">and</span> <span m="204760">1/2.</span> <span m="205320">And</span> <span m="205480">as</span> <span m="205560">you</span> <span m="205650">can</span> <span m="205790">see</span> <span m="206000">from</span> <span m="206150">this</span> <span m="206310">example,</span> <span m="207880">the</span> <span m="207920">expected</span> <span m="208550">value</span> <span m="209060">doesn't</span> <span m="209590">have</span> <span m="209980">to</span> <span m="210090">be</span> <span m="210280">attainable</span> <span m="210840">by</span> <span m="210990">one</span> <span m="211210">of</span> <span m="211280">the</span> <span m="211390">outcomes.</span> <span m="213030">You</span> <span m="213150">don't</span> <span m="213820">get</span> <span m="214090">a</span> <span m="214130">3</span> <span m="214370">and</span> <span m="214610">1/2</span> <span m="214850">when</span> <span m="214950">you</span> <span m="215020">roll</span> <span m="215270">a</span> <span m="215320">die.</span> <span m="216730">But</span> <span m="216840">that's</span> <span m="217060">the</span> <span m="217200">average</span> <span m="217540">value.</span> </p>
<p><span m="219840">Now</span> <span m="220150">the</span> <span m="221210">expected</span> <span m="221800">value,</span> <span m="222380">the</span> <span m="222490">expectation,</span> <span m="223330">the</span> <span m="223470">average,</span> <span m="224080">the</span> <span m="224160">mean,</span> <span m="224610">they're</span> <span m="224760">all</span> <span m="225100">the</span> <span m="225180">same</span> <span m="225550">thing.</span> <span m="227000">They're</span> <span m="227110">all</span> <span m="227320">defined</span> <span m="227770">based</span> <span m="228040">on a</span> <span m="228190">random</span> <span m="228470">variable.</span> <span m="228770">They</span> <span m="228880">all</span> <span m="229060">mean</span> <span m="229240">the</span> <span m="229300">same</span> <span m="229590">thing.</span> <span m="230890">They</span> <span m="230980">are</span> <span m="231070">all</span> <span m="231260">different</span> <span m="231890">than</span> <span m="232800">the</span> <span m="232920">median.</span> <span m="234650">That's</span> <span m="234870">something</span> <span m="235090">that's</span> <span m="235260">different.</span> <span m="235800">So</span> <span m="235830">let</span> <span m="235920">me</span> <span m="236690">define</span> <span m="237170">the</span> <span m="237230">median.</span> </p>
<p><span m="238400">Basically</span> <span m="239130">it</span> <span m="239230">is</span> <span m="239430">the</span> <span m="240050">outcome</span> <span m="240690">which</span> <span m="241250">splits</span> <span m="242090">the</span> <span m="242180">probabilities</span> <span m="242850">in</span> <span m="242960">half.</span> <span m="243390">In</span> <span m="243480">other</span> <span m="243640">words,</span> <span m="243820">you've</span> <span m="243920">got</span> <span m="244040">a</span> <span m="244070">50%</span> <span m="244760">chance</span> <span m="245030">of</span> <span m="245120">being</span> <span m="245490">bigger</span> <span m="245790">than</span> <span m="245920">the</span> <span m="245980">median</span> <span m="246880">and</span> <span m="247030">a</span> <span m="247060">50%</span> <span m="247690">chance</span> <span m="247970">of</span> <span m="248050">being</span> <span m="248260">smaller.</span> <span m="249760">Now</span> <span m="249950">precisely</span> <span m="253440">you</span> <span m="253610">can</span> <span m="253730">define</span> <span m="254050">it</span> <span m="254150">this</span> <span m="254360">way.</span> <span m="257209">The</span> <span m="257339">median</span> <span m="261850">of</span> <span m="262019">a</span> <span m="262110">random</span> <span m="262450">variable</span> <span m="262890">R</span> <span m="264950">is</span> <span m="265200">the</span> <span m="265280">value</span> <span m="265910">in</span> <span m="265990">the</span> <span m="266090">range</span> <span m="266500">of</span> <span m="266630">R</span> <span m="271190">such</span> <span m="271560">that</span> <span m="273750">the</span> <span m="273860">probability</span> <span m="275300">that</span> <span m="275490">the</span> <span m="275600">random</span> <span m="275960">variable</span> <span m="276500">is</span> <span m="276760">less</span> <span m="277270">than</span> <span m="277430">this</span> <span m="278140">median</span> <span m="278570">point</span> <span m="279430">is</span> <span m="279700">at</span> <span m="279790">most</span> <span m="280150">1/2.</span> <span m="282020">And</span> <span m="282290">the</span> <span m="282360">probability</span> <span m="283440">that</span> <span m="283590">random</span> <span m="283900">variable</span> <span m="284330">is</span> <span m="284460">bigger</span> <span m="285270">than</span> <span m="285440">the</span> <span m="285500">median</span> <span m="286390">is</span> <span m="286510">strictly</span> <span m="287080">less</span> <span m="287410">than</span> <span m="287600">1/2.</span> </p>
<p><span m="290910">Now,</span> <span m="291390">some</span> <span m="291750">texts</span> <span m="292120">do it</span> <span m="292490">differently.</span> <span m="292950">They</span> <span m="293070">swap</span> <span m="294430">this</span> <span m="295230">less</span> <span m="295510">than</span> <span m="295660">or</span> <span m="295750">equal</span> <span m="295960">to</span> <span m="296120">with</span> <span m="296260">this</span> <span m="296470">one.</span> <span m="297370">And</span> <span m="297510">it</span> <span m="297540">could</span> <span m="297680">give</span> <span m="297860">you</span> <span m="297950">a</span> <span m="298000">different</span> <span m="298320">answer</span> <span m="298610">if</span> <span m="298700">you</span> <span m="298800">do</span> <span m="298990">that.</span> <span m="301170">And</span> <span m="301430">I</span> <span m="301480">think</span> <span m="301720">in</span> <span m="301800">the</span> <span m="301880">text,</span> <span m="302380">actually,</span> <span m="303550">we</span> <span m="303710">screwed</span> <span m="304000">it</span> <span m="304110">up.</span> <span m="304820">So</span> <span m="305030">we</span> <span m="305110">have</span> <span m="305220">to</span> <span m="305280">get</span> <span m="305390">that</span> <span m="305570">corrected.</span> <span m="305880">I</span> <span m="305960">think</span> <span m="306210">we</span> <span m="306280">might</span> <span m="306500">have</span> <span m="306600">put</span> <span m="306770">a</span> <span m="306820">less</span> <span m="307080">than</span> <span m="307210">or</span> <span m="307290">equal</span> <span m="307530">here,</span> <span m="307740">which</span> <span m="308020">is</span> <span m="308170">not</span> <span m="308430">the</span> <span m="308520">right</span> <span m="308830">definition.</span> <span m="309990">So</span> <span m="310140">this</span> <span m="310410">is</span> <span m="310670">the</span> <span m="310760">right</span> <span m="310990">definition</span> <span m="311470">that</span> <span m="311570">we'll</span> <span m="311710">use.</span> </p>
<p><span m="313160">So</span> <span m="313390">using</span> <span m="313750">this</span> <span m="313910">definition,</span> <span m="314650">what</span> <span m="314910">is</span> <span m="315150">the</span> <span m="315250">median</span> <span m="316730">of</span> <span m="317030">the</span> <span m="317140">random</span> <span m="317500">variable</span> <span m="317880">that</span> <span m="317980">corresponds</span> <span m="318650">to</span> <span m="318750">rolling</span> <span m="319070">a</span> <span m="319140">die?</span> <span m="320950">What's</span> <span m="321220">the</span> <span m="321300">median</span> <span m="322680">when</span> <span m="322830">I</span> <span m="322910">roll</span> <span m="323180">a</span> <span m="323250">six</span> <span m="323510">sided</span> <span m="323850">fair</span> <span m="324110">die?</span> <span m="327810">Not</span> <span m="328080">3</span> <span m="328250">and</span> <span m="328390">1/2,</span> <span m="329545">because</span> <span m="329930">it's</span> <span m="330080">got</span> <span m="330280">to</span> <span m="330350">be</span> <span m="330440">one</span> <span m="330600">of</span> <span m="330660">the</span> <span m="330730">values.</span> <span m="332820">The</span> <span m="332920">median</span> <span m="333360">does</span> <span m="334020">have</span> <span m="334230">to</span> <span m="334340">be</span> <span m="334450">in</span> <span m="334593">the</span> <span m="334736">realm</span> <span m="335740">of</span> <span m="335900">the</span> <span m="335980">random</span> <span m="336270">variable.</span> <span m="336610">It has</span> <span m="336810">to</span> <span m="336960">one</span> <span m="337090">of</span> <span m="337150">the</span> <span m="337520">attainable</span> <span m="337820">values.</span> <span m="339340">What's</span> <span m="339570">the</span> <span m="339660">median</span> <span m="340110">value</span> <span m="340460">when</span> <span m="340590">I</span> <span m="340660">roll</span> <span m="340900">a</span> <span m="340950">die?</span> </p>
<p><span m="341900">AUDIENCE: Four.</span> </p>
<p><span m="343280">PROFESSOR: Four.</span> <span m="343980">Let's</span> <span m="344150">try</span> <span m="344420">that</span> <span m="344670">out.</span> <span m="345580">If I</span> <span m="345700">plug in</span> <span m="346220">4</span> <span m="346740">here,</span> <span m="347690">the</span> <span m="347800">probability</span> <span m="348360">I'm</span> <span m="348510">less</span> <span m="348850">than</span> <span m="349010">4</span> <span m="349620">is</span> <span m="349940">1/2.</span> <span m="350790">Could</span> <span m="350920">be</span> <span m="351000">1,</span> <span m="351240">2,</span> <span m="351400">or</span> <span m="351460">3.</span> <span m="352540">The</span> <span m="352660">probability</span> <span m="353160">of</span> <span m="353250">greater</span> <span m="353730">than</span> <span m="353910">4</span> <span m="355150">is</span> <span m="355520">5</span> <span m="355790">or</span> <span m="355880">six</span> <span m="356460">is</span> <span m="356600">1/3.</span> <span m="357330">That's</span> <span m="357610">less</span> <span m="357830">than</span> <span m="357940">1/2.</span> <span m="358470">So</span> <span m="358660">4</span> <span m="359020">works.</span> <span m="360280">3</span> <span m="361900">doesn't</span> <span m="362330">work,</span> <span m="363390">because</span> <span m="363570">the</span> <span m="363650">probability</span> <span m="364160">I'm</span> <span m="364290">bigger</span> <span m="364530">than</span> <span m="364670">3</span> <span m="365290">is</span> <span m="365650">1/2.</span> <span m="366710">So</span> <span m="366850">it</span> <span m="366920">doesn't</span> <span m="367220">work.</span> <span m="368320">And</span> <span m="368490">the</span> <span m="368590">other</span> <span m="368760">definition</span> <span m="369270">sometimes</span> <span m="369610">people</span> <span m="369860">use</span> <span m="370170">would</span> <span m="370310">turn</span> <span m="370530">out</span> <span m="370840">that</span> <span m="370980">three</span> <span m="371320">is</span> <span m="371490">the</span> <span m="371570">median.</span> </p>
<p><span m="374560">Now,</span> <span m="374800">we're</span> <span m="374980">not</span> <span m="375170">going</span> <span m="375290">to</span> <span m="375380">spend</span> <span m="375810">any</span> <span m="375960">more</span> <span m="376160">time</span> <span m="376520">talking</span> <span m="376850">about</span> <span m="377030">the</span> <span m="377100">median.</span> <span m="378330">What's</span> <span m="378540">really</span> <span m="378800">important in</span> <span m="379290">probability</span> <span m="379930">is</span> <span m="380070">the</span> <span m="380160">mean.</span> <span m="381430">And</span> <span m="381560">that's</span> <span m="381690">because</span> <span m="381870">you</span> <span m="381960">can</span> <span m="382110">do</span> <span m="382300">a</span> <span m="382440">whole</span> <span m="382900">lot</span> <span m="383170">more</span> <span m="383390">with</span> <span m="383620">it</span> <span m="383940">in</span> <span m="384110">applications.</span> <span m="385970">Any</span> <span m="386190">question</span> <span m="386580">about</span> <span m="386780">the</span> <span m="386850">definitions</span> <span m="389550">so</span> <span m="389710">far?</span> </p>
<p><span m="391770">All</span> <span m="391990">right,</span> <span m="392690">so</span> <span m="393810">now</span> <span m="394010">we're</span> <span m="394070">going</span> <span m="394200">to</span> <span m="394260">do</span> <span m="394370">a</span> <span m="394410">more</span> <span m="394750">interesting</span> <span m="395150">example.</span> <span m="396890">We're</span> <span m="397100">going</span> <span m="397230">to</span> <span m="397790">play</span> <span m="398150">a</span> <span m="398230">gambling</span> <span m="398770">game</span> <span m="399780">and</span> <span m="399960">we're</span> <span m="400040">going</span> <span m="400160">to</span> <span m="400220">analyze</span> <span m="401160">the</span> <span m="401300">expected</span> <span m="401930">winnings,</span> <span m="402430">the</span> <span m="402540">expected</span> <span m="403110">return.</span> <span m="404810">And</span> <span m="405270">this</span> <span m="405430">is</span> <span m="405530">a</span> <span m="405740">simple</span> <span m="406160">three</span> <span m="406380">person</span> <span m="406760">game</span> <span m="407090">that</span> <span m="407210">you</span> <span m="407320">can</span> <span m="407630">see</span> <span m="407830">played</span> <span m="408740">in</span> <span m="408930">bars</span> <span m="410460">and</span> <span m="410960">informally.</span> <span m="412480">But</span> <span m="412610">to</span> <span m="412700">play</span> <span m="413030">it</span> <span m="413120">I</span> <span m="413210">need</span> <span m="413420">a</span> <span m="413450">volunteer</span> <span m="415590">from</span> <span m="415810">the</span> <span m="415880">class,</span> <span m="416340">somebody</span> <span m="416630">who</span> <span m="416720">wants</span> <span m="416970">to</span> <span m="417070">play.</span> <span m="417640">You've</span> <span m="417920">done</span> <span m="418160">it.</span> <span m="418250">Who</span> <span m="418360">hasn't</span> <span m="418640">done</span> <span m="418740">it?</span> <span m="418820">Have you</span> <span m="418920">done</span> <span m="419100">it?</span> <span m="420500">You</span> <span m="420640">haven't</span> <span m="420880">played.</span> <span m="421150">Do</span> <span m="421500">you</span> <span m="421570">have</span> <span m="421670">money</span> <span m="421890">in</span> <span m="421950">your</span> <span m="422050">pocket?</span> <span m="423250">You</span> <span m="423350">got</span> <span m="423450">to</span> <span m="423520">borrow</span> <span m="423770">some</span> <span m="423940">money.</span> <span m="424220">If</span> <span m="424750">you</span> <span m="424810">don't have</span> <span m="424910">any</span> <span m="425030">money,</span> <span m="425250">you got</span> <span m="425380">to</span> <span m="425440">borrow</span> <span m="425740">some.</span> </p>
<p><span m="427048">AUDIENCE: How</span> <span m="427484">much</span> <span m="427920">money?</span> </p>
<p><span m="428770">PROFESSOR: Oh,</span> <span m="429060">I</span> <span m="429160">don't</span> <span m="429270">know,</span> <span m="429650">$5 or</span> <span m="429700">$10.</span> <span m="431560">Got</span> <span m="431860">it?</span> <span m="432120">All right,</span> <span m="432750">come</span> <span m="432920">on</span> <span m="433080">down.</span> <span m="436020">you're</span> <span m="436140">going</span> <span m="436260">to</span> <span m="436330">play</span> <span m="436530">against</span> <span m="436810">a</span> <span m="436840">couple</span> <span m="437070">of</span> <span m="437140">TAs.</span> <span m="437620">Maybe</span> <span m="437830">a</span> <span m="438370">couple</span> <span m="438600">of</span> <span m="438680">you</span> <span m="438780">will want</span> <span m="438960">to</span> <span m="439030">play.</span> <span m="440910">I'll</span> <span m="441200">get</span> <span m="441410">Nick and</span> <span m="441760">Martyna</span> <span m="442180">to</span> <span m="442280">play</span> <span m="442590">here.</span> </p>
<p><span m="443630">Now,</span> <span m="444410">this</span> <span m="444630">is</span> <span m="444750">a</span> <span m="444800">very</span> <span m="445330">simple</span> <span m="445780">game.</span> <span m="447960">In</span> <span m="448130">each</span> <span m="448450">round,</span> <span m="449220">each</span> <span m="449520">of</span> <span m="449620">the</span> <span m="449710">players</span> <span m="450380">is</span> <span m="450500">going</span> <span m="450630">to</span> <span m="450710">wager</span> <span m="451280">$2.</span> <span m="453060">So</span> <span m="454020">I</span> <span m="454210">can</span> <span m="454370">loan</span> <span m="454620">you</span> <span m="454710">guys</span> <span m="454970">some</span> <span m="455110">money</span> <span m="455340">here,</span> <span m="455510">I</span> <span m="455590">guess.</span> <span m="456950">So</span> <span m="457090">you</span> <span m="457150">have</span> <span m="457260">$2?</span> <span m="460120">All</span> <span m="460250">right,</span> <span m="460370">I have</span> <span m="460550">to</span> <span m="460600">loan</span> <span m="460840">you</span> <span m="460920">money.</span> <span m="461240">I</span> <span m="461310">better</span> <span m="461500">count</span> <span m="461770">this</span> <span m="462208">out.</span> </p>
<p><span m="462646">AUDIENCE: I</span> <span m="462755">think</span> <span m="462865">I can</span> <span m="462974">find</span> <span m="463084">the Zimbabwe</span> <span m="463522">dollars</span> <span m="463960">I had</span> <span m="464400">here</span> <span m="465400">somewhere.</span> </p>
<p><span m="465900">PROFESSOR: I</span> <span m="466100">don't</span> <span m="466270">know</span> <span m="466400">if</span> <span m="466490">I'm</span> <span m="466670">taking</span> <span m="467060">those.</span> <span m="467260">Here,</span> <span m="467410">you</span> <span m="467470">can</span> <span m="467600">split</span> <span m="467850">that</span> <span m="468060">up.</span> <span m="468390">I</span> <span m="468610">loaned</span> <span m="468920">you</span> <span m="469000">guys</span> <span m="469220">$10</span> <span m="469700">there.</span> <span m="469850">You</span> <span m="469910">got</span> <span m="470040">$2.</span> <span m="470880">Put</span> <span m="471060">it</span> <span m="471140">on</span> <span m="471230">the</span> <span m="471300">table.</span> <span m="473780">Wager your</span> <span m="474140">$2.</span> <span m="475760">Now</span> <span m="476100">what</span> <span m="476350">we're</span> <span m="476440">going</span> <span m="476570">to</span> <span m="476640">do</span> <span m="477180">is</span> <span m="477430">they're</span> <span m="477840">each</span> <span m="478160">going</span> <span m="478290">to</span> <span m="478360">guess</span> <span m="479440">the</span> <span m="479590">results</span> <span m="480050">of</span> <span m="480120">a</span> <span m="480170">coin</span> <span m="480480">toss.</span> <span m="482230">We'll</span> <span m="482380">put</span> <span m="482570">the pot</span> <span m="482650">together</span> <span m="483530">here,</span> <span m="484060">mix</span> <span m="484300">it</span> <span m="484380">up.</span> <span m="485600">You only</span> <span m="485730">have</span> <span m="485880">to</span> <span m="485950">do</span> <span m="486070">two.</span> <span m="486370">You</span> <span m="486470">can</span> <span m="486650">save</span> <span m="486950">those</span> <span m="487150">for</span> <span m="487250">now.</span> <span m="488250">Oh,</span> <span m="488370">that's</span> <span m="488540">your</span> <span m="488670">two.</span> </p>
<p><span m="489420">All</span> <span m="489670">right,</span> <span m="489940">so</span> <span m="490070">we</span> <span m="490140">have</span> <span m="490270">$6</span> <span m="490920">in</span> <span m="490980">the</span> <span m="491050">pot.</span> <span m="492220">Now,</span> <span m="493070">each</span> <span m="493380">of</span> <span m="493510">you</span> <span m="493970">is</span> <span m="494080">going</span> <span m="494200">to</span> <span m="494270">guess</span> <span m="494600">the</span> <span m="494720">outcome</span> <span m="495030">of</span> <span m="495110">a</span> <span m="495150">coin.</span> <span m="495670">So</span> <span m="495840">here's</span> <span m="496770">the</span> <span m="496880">guesses.</span> <span m="497330">Heads</span> <span m="497600">or</span> <span m="497660">tails,</span> <span m="498180">that's</span> <span m="498250">for</span> <span m="498390">you.</span> <span m="499930">Heads</span> <span m="500250">or</span> <span m="500320">tails,</span> <span m="500830">the</span> <span m="500910">guess</span> <span m="501220">for</span> <span m="501350">you.</span> <span m="501710">And</span> <span m="501830">you</span> <span m="501920">get</span> <span m="502080">a</span> <span m="502140">heads</span> <span m="502360">or</span> <span m="502410">a</span> <span m="502510">tails.</span> <span m="502690">So</span> <span m="502760">instead</span> <span m="502970">of</span> <span m="503050">writing</span> <span m="503380">it</span> <span m="503480">down,</span> <span m="504220">you're</span> <span m="504270">going</span> <span m="504390">to</span> <span m="504450">sort</span> <span m="504590">of</span> <span m="504660">hide</span> <span m="505060">those,</span> <span m="506180">maybe</span> <span m="506430">get</span> <span m="506590">some</span> <span m="506700">advice</span> <span m="507070">from</span> <span m="507140">the</span> <span m="507210">class,</span> <span m="507640">and</span> <span m="507730">you're</span> <span m="507820">going</span> <span m="507940">to</span> <span m="508000">pick</span> <span m="508270">one</span> <span m="509450">and</span> <span m="509630">put</span> <span m="509800">it</span> <span m="510200">on</span> <span m="510570">the</span> <span m="510640">table</span> <span m="511000">here</span> <span m="511230">as</span> <span m="511320">your</span> <span m="511510">guess.</span> </p>
<p><span m="512409">And</span> <span m="512610">then</span> <span m="512760">one</span> <span m="513020">of</span> <span m="513130">you</span> <span m="513440">out</span> <span m="513650">there</span> <span m="513890">is</span> <span m="513970">going</span> <span m="514090">to</span> <span m="514150">flip</span> <span m="514440">a</span> <span m="514490">coin</span> <span m="515650">and</span> <span m="515809">announce</span> <span m="516870">the</span> <span m="517000">result.</span> <span m="518460">And</span> <span m="518659">then</span> <span m="518809">we're</span> <span m="518909">going</span> <span m="519030">to</span> <span m="519590">reveal</span> <span m="520169">their</span> <span m="520320">guesses</span> <span m="521080">and</span> <span m="521370">the</span> <span m="521440">winners</span> <span m="522480">split</span> <span m="523000">the</span> <span m="523070">money.</span> <span m="524670">And</span> <span m="524870">if</span> <span m="525010">there's</span> <span m="525370">no</span> <span m="525630">winners,</span> <span m="526470">well,</span> <span m="526700">they</span> <span m="526860">take</span> <span m="527110">their</span> <span m="527210">$2</span> <span m="527740">back</span> <span m="527940">and</span> <span m="528030">they</span> <span m="528100">split</span> <span m="528400">the</span> <span m="528480">pot.</span> </p>
<p><span m="529570">So</span> <span m="529710">this</span> <span m="529880">is</span> <span m="529980">a</span> <span m="530020">very</span> <span m="530450">fair</span> <span m="530830">game.</span> <span m="532520">They</span> <span m="532720">make</span> <span m="532940">their</span> <span m="533040">choices,</span> <span m="534150">you</span> <span m="534460">guys</span> <span m="534810">toss</span> <span m="535130">the</span> <span m="535220">coin,</span> <span m="536200">and</span> <span m="536380">the</span> <span m="536460">winners</span> <span m="536770">share</span> <span m="536930">of</span> <span m="536990">the</span> <span m="537050">pot.</span> <span m="537350">If</span> <span m="537440">there's</span> <span m="537620">one,</span> <span m="537910">winner</span> <span m="538190">they</span> <span m="538300">get</span> <span m="539380">$6,</span> <span m="540050">which</span> <span m="540200">is</span> <span m="540280">a</span> <span m="540330">profit</span> <span m="540790">of $4.</span> <span m="541650">Two</span> <span m="541860">winners,</span> <span m="542310">they</span> <span m="542400">each</span> <span m="542580">get</span> <span m="542750">three,</span> <span m="543310">profit</span> <span m="543680">of</span> <span m="543760">one,</span> <span m="544480">the</span> <span m="544620">other</span> <span m="544770">one's</span> <span m="544980">out</span> <span m="545210">two</span> <span m="545360">bucks.</span> <span m="546790">Nobody</span> <span m="547160">wins,</span> <span m="547470">they</span> <span m="547550">get</span> <span m="547660">their</span> <span m="547760">money</span> <span m="548010">back,</span> <span m="548240">profit</span> <span m="548620">0.</span> <span m="548940">They</span> <span m="549070">all</span> <span m="549330">win,</span> <span m="549550">they</span> <span m="549640">get</span> <span m="549780">their</span> <span m="549890">money</span> <span m="550110">back,</span> <span m="550340">profit</span> <span m="550690">0.</span> </p>
<p><span m="551300">Is</span> <span m="551410">the game</span> <span m="551770">clear?</span> <span m="553520">Who's</span> <span m="553700">got</span> <span m="553870">a</span> <span m="553900">coin</span> <span m="554270">out</span> <span m="554470">there?</span> <span m="555810">You've</span> <span m="555960">got</span> <span m="556060">a</span> <span m="556090">coin?</span> <span m="557030">Very</span> <span m="557390">good.</span> <span m="557870">Now</span> <span m="558390">you've</span> <span m="558590">got</span> <span m="558730">to</span> <span m="558790">guess</span> <span m="559060">the</span> <span m="559140">coin.</span> <span m="559510">Don't</span> <span m="559940">show</span> <span m="560240">each</span> <span m="560430">other</span> <span m="560580">your</span> <span m="560710">guesses.</span> <span m="562520">And</span> <span m="562740">put</span> <span m="563020">your</span> <span m="563160">guesses</span> <span m="563830">on</span> <span m="564180">the</span> <span m="564820">table</span> <span m="565230">here.</span> <span m="566490">You</span> <span m="566620">get</span> <span m="566740">to</span> <span m="566820">pick</span> <span m="567070">heads</span> <span m="567330">or</span> <span m="567390">tails.</span> <span m="567750">Put</span> <span m="567930">it on</span> <span m="567990">the</span> <span m="568060">table.</span> </p>
<p><span m="575270">[LAUGHS]</span> </p>
<p><span m="576620">This</span> <span m="576810">is</span> <span m="576920">not</span> <span m="577130">a</span> <span m="577190">hard</span> <span m="577450">game.</span> <span m="581140">You</span> <span m="581270">got</span> <span m="581420">a</span> <span m="581450">heads</span> <span m="581700">and</span> <span m="581790">a</span> <span m="581820">tails,</span> <span m="582280">right?</span> <span m="584710">Well,</span> <span m="584890">yeah,</span> <span m="585010">did</span> <span m="585090">you</span> <span m="585480">put</span> <span m="585650">a</span> <span m="585730">heads</span> <span m="585930">or a</span> <span m="586000">tails</span> <span m="586290">in</span> <span m="586350">there?</span> <span m="588198">Yeah,</span> <span m="589110">there's</span> <span m="589280">one</span> <span m="589550">heads,</span> <span m="589900">one</span> <span m="590140">tails.</span> <span m="591650">No,</span> <span m="591770">that's</span> <span m="591990">blank.</span> <span m="592420">You</span> <span m="592550">don't</span> <span m="592740">win</span> <span m="592980">that</span> <span m="593220">way.</span> <span m="593950">There</span> <span m="594200">you</span> <span m="594300">go.</span> <span m="594450">That's</span> <span m="594670">the</span> <span m="594780">tails.</span> </p>
<p><span m="595100">AUDIENCE: I thought</span> <span m="595420">you</span> <span m="595715">meant--</span> </p>
<p><span m="596940">PROFESSOR: Heads.</span> <span m="597690">So</span> <span m="597800">if</span> <span m="597940">you</span> <span m="598030">want</span> <span m="598220">heads,</span> <span m="598530">you</span> <span m="598640">go</span> <span m="598800">like</span> <span m="598990">that.</span> </p>
<p><span m="599418">AUDIENCE: Yeah.</span> <span m="599846">I</span> <span m="599953">thought</span> <span m="600060">you</span> <span m="600167">meant</span> <span m="600274">do</span> <span m="600702">it the other</span> <span m="601130">way.</span> </p>
<p><span m="601560">PROFESSOR: There</span> <span m="601780">you</span> <span m="601880">go.</span> </p>
<p><span m="602345">AUDIENCE: All right,</span> <span m="602810">I'll have to make</span> <span m="603275">these</span> <span m="603740">indistinguishable</span> <span m="604205">now.</span> </p>
<p><span m="605140">PROFESSOR: Well,</span> <span m="605940">yeah.</span> <span m="606140">One</span> <span m="606320">of</span> <span m="606380">them</span> <span m="606470">is</span> <span m="606550">all</span> <span m="606680">ripped</span> <span m="606920">up</span> <span m="607070">now.</span> <span m="609030">They</span> <span m="609250">already</span> <span m="609520">guessed,</span> <span m="610210">so</span> <span m="610350">they've</span> <span m="610570">already</span> <span m="610820">done</span> <span m="611030">their</span> <span m="611160">guess.</span> <span m="611490">So</span> <span m="611590">you</span> <span m="611700">pick</span> <span m="611940">one</span> <span m="612110">and</span> <span m="612200">put</span> <span m="612360">it</span> <span m="612450">there.</span> </p>
<p><span m="612800">AUDIENCE: This is what I</span> <span m="613212">was originally</span> <span m="613624">going to guess</span> <span m="614036">anyway.</span> </p>
<p><span m="616030">PROFESSOR: So</span> <span m="616200">that's</span> <span m="616480">good.</span> <span m="617080">And</span> <span m="618190">you</span> <span m="618400">now</span> <span m="619310">flip</span> <span m="619990">the</span> <span m="620120">coin</span> <span m="622160">and</span> <span m="622640">tell</span> <span m="622880">us</span> <span m="623010">what</span> <span m="623140">it</span> <span m="623210">came</span> <span m="623450">out.</span> </p>
<p><span m="624690">AUDIENCE: Heads.</span> </p>
<p><span m="625690">PROFESSOR: Heads.</span> <span m="626270">All right,</span> <span m="626440">let's</span> <span m="626660">reveal</span> <span m="627130">your</span> <span m="627270">choices.</span> <span m="630280">All</span> <span m="630530">right,</span> <span m="630870">so</span> <span m="631020">there's</span> <span m="631190">one</span> <span m="631530">heads.</span> <span m="631930">Martyna</span> <span m="632470">takes</span> <span m="632850">all</span> <span m="633340">the</span> <span m="633430">money.</span> <span m="635780">Just</span> <span m="635960">bad</span> <span m="636290">luck</span> <span m="636510">there,</span> <span m="636740">right?</span> <span m="637380">We're</span> <span m="637470">going</span> <span m="637590">to</span> <span m="637650">play</span> <span m="637880">again.</span> <span m="638460">We're</span> <span m="638520">going</span> <span m="638640">to</span> <span m="638700">play</span> <span m="638860">a</span> <span m="638910">few</span> <span m="639070">times.</span> <span m="639400">But</span> <span m="639500">let's</span> <span m="639540">start</span> <span m="639780">recording</span> <span m="640940">what</span> <span m="641280">happened</span> <span m="641700">here.</span> <span m="643490">So</span> <span m="643680">the</span> <span m="643790">coin</span> <span m="644450">came</span> <span m="644770">out</span> <span m="645360">heads.</span> <span m="646850">OK,</span> <span m="647130">remind</span> <span m="647400">me</span> <span m="647480">your</span> <span m="647590">name?</span> </p>
<p><span m="648005">AUDIENCE: Adam.</span> </p>
<p><span m="648420">PROFESSOR: Adam</span> <span m="649360">gassed</span> <span m="650470">tails.</span> <span m="654850">And</span> <span m="655620">Martyna.</span> <span m="659230">Is</span> <span m="659350">it</span> <span m="659450">Y?</span> <span m="661290">Martyna</span> <span m="662110">guessed</span> <span m="662500">heads</span> <span m="663950">and</span> <span m="664230">Nick,</span> <span m="666020">he</span> <span m="666220">also</span> <span m="666520">loses</span> <span m="666890">at</span> <span m="667020">tails.</span> <span m="667680">So</span> <span m="667760">in</span> <span m="667930">this</span> <span m="668160">case</span> <span m="669100">Adam</span> <span m="670330">is</span> <span m="670730">out</span> <span m="671050">$2.</span> <span m="673780">Martyna</span> <span m="674340">got</span> <span m="674510">a</span> <span m="674550">profit</span> <span m="675030">of</span> <span m="675120">$4.</span> <span m="676800">And</span> <span m="676970">Nick</span> <span m="677180">is</span> <span m="677300">also</span> <span m="677610">down</span> <span m="678000">$2.</span> </p>
<p><span m="680230">All</span> <span m="680510">right,</span> <span m="680840">let's</span> <span m="681030">try</span> <span m="681320">it</span> <span m="681410">again.</span> <span m="682800">So</span> <span m="683100">again,</span> <span m="683610">pick</span> <span m="683810">another</span> <span m="684430">heads</span> <span m="684710">or</span> <span m="684770">tails.</span> <span m="686500">Adam's</span> <span m="686910">busily</span> <span m="687300">working</span> <span m="687660">out</span> <span m="687770">probabilities</span> <span m="688450">here.</span> <span m="691370">Well you</span> <span m="691500">got</span> <span m="691610">to</span> <span m="691680">put</span> <span m="691800">your</span> <span m="691890">money</span> <span m="692090">on</span> <span m="692150">the</span> <span m="692220">table</span> <span m="692510">here</span> <span m="692690">now.</span> <span m="693010">You</span> <span m="693090">got</span> <span m="693240">$2.</span> <span m="695700">That's</span> <span m="695910">three.</span> <span m="697480">Here</span> <span m="697650">you</span> <span m="697720">go.</span> </p>
<p><span m="698517">[LAUGHS]</span> </p>
<p><span m="700010">There</span> <span m="700290">we</span> <span m="700350">go.</span> <span m="700490">Two</span> <span m="700650">more</span> <span m="700810">dollars.</span> <span m="701330">You</span> <span m="701440">can</span> <span m="701640">make</span> <span m="701850">change</span> <span m="702130">if</span> <span m="702240">you</span> <span m="702330">want.</span> <span m="703390">There</span> <span m="703680">we</span> <span m="703800">go.</span> <span m="704650">All</span> <span m="704760">right,</span> <span m="705210">make</span> <span m="705380">your</span> <span m="705470">choices.</span> <span m="707990">You</span> <span m="708170">got</span> <span m="708400">one?</span> <span m="710800">As</span> <span m="710980">long</span> <span m="711200">as</span> <span m="711340">you</span> <span m="711410">guys</span> <span m="711580">don't</span> <span m="711760">show</span> <span m="712140">him</span> <span m="712500">your</span> <span m="712600">choices,</span> <span m="713050">we</span> <span m="713160">should</span> <span m="713380">be</span> <span m="713490">good</span> <span m="713720">here,</span> <span m="713910">I</span> <span m="714000">think.</span> <span m="715340">And</span> <span m="715800">you</span> <span m="715860">got</span> <span m="715970">to</span> <span m="716040">pick</span> <span m="716240">one</span> <span m="716420">here.</span> <span m="720150">Very</span> <span m="720420">good.</span> <span m="720640">Can we</span> <span m="720770">have</span> <span m="720860">a</span> <span m="720910">coin</span> <span m="721210">toss</span> <span m="721510">please?</span> </p>
<p><span m="723370">AUDIENCE: Heads.</span> </p>
<p><span m="724300">PROFESSOR: Heads</span> <span m="724770">again.</span> <span m="725460">What</span> <span m="725470">do</span> <span m="725530">we</span> <span m="725620">got?</span> <span m="727560">All</span> <span m="727680">right,</span> <span m="727940">good</span> <span m="728130">job.</span> <span m="728480">OK,</span> <span m="728780">Adam</span> <span m="729100">got</span> <span m="729320">heads</span> <span m="729725">and</span> <span m="730130">Martyna's</span> <span m="730810">on</span> <span m="730910">a</span> <span m="731000">roll</span> <span m="731310">here.</span> <span m="731590">So</span> <span m="731690">we</span> <span m="731750">got</span> <span m="731840">to</span> <span m="731900">split</span> <span m="732190">this</span> <span m="732420">up.</span> <span m="732630">Can</span> <span m="732760">you</span> <span m="732850">make</span> <span m="733470">change</span> <span m="733890">for</span> <span m="734110">him</span> <span m="734210">here?</span> <span m="734360">He</span> <span m="734440">gets</span> <span m="734670">$3.</span> <span m="736080">So</span> <span m="736980">it</span> <span m="737070">was</span> <span m="737230">heads.</span> <span m="738750">Adam</span> <span m="739150">had</span> <span m="739420">heads,</span> <span m="740310">Martyna</span> <span m="740770">has</span> <span m="741090">heads,</span> <span m="742480">and</span> <span m="742690">Nick's</span> <span m="743320">in</span> <span m="743470">trouble.</span> <span m="743870">Nick's</span> <span m="744190">off</span> <span m="744460">another</span> <span m="744750">$2</span> <span m="745930">of</span> <span m="746090">my</span> <span m="746410">money.</span> <span m="748180">Martyna</span> <span m="748780">is</span> <span m="749090">up</span> <span m="749510">$1</span> <span m="750400">because</span> <span m="750620">she</span> <span m="750770">wagered</span> <span m="751220">two</span> <span m="751520">and</span> <span m="751650">got</span> <span m="751850">three</span> <span m="752690">and</span> <span m="752910">Adam</span> <span m="753990">is</span> <span m="754590">up</span> <span m="754770">$1</span> <span m="755220">here</span> <span m="755920">for</span> <span m="756050">that one.</span> </p>
<p><span m="757570">All</span> <span m="757800">right,</span> <span m="758190">let's</span> <span m="758440">make</span> <span m="758660">another</span> <span m="758920">selection.</span> <span m="759420">Heads</span> <span m="759650">or</span> <span m="759710">tails?</span> <span m="761700">Put</span> <span m="761880">the</span> <span m="761950">money</span> <span m="762260">up</span> <span m="762380">here,</span> <span m="762730">$2.</span> <span m="763860">Going</span> <span m="764220">to</span> <span m="764330">need</span> <span m="764490">another</span> <span m="764760">loan</span> <span m="765010">here,</span> <span m="765250">Nick?</span> </p>
<p><span m="767786">AUDIENCE:  I can throw in</span> <span m="768192">my keys.</span> </p>
<p><span m="769410">PROFESSOR: No, no.</span> </p>
<p><span m="770910">AUDIENCE: The</span> <span m="771410">keys to</span> <span m="771905">my</span> <span m="772400">non-existent</span> <span m="772895">car.</span> </p>
<p><span m="773390">PROFESSOR: You</span> <span m="773550">got</span> <span m="773650">money?</span> <span m="773930">OK.</span> <span m="774610">All</span> <span m="774880">right.</span> </p>
<p><span m="775335">AUDIENCE: I</span> <span m="775410">got</span> <span m="775486">a</span> <span m="775562">lot</span> <span m="775638">of</span> <span m="775714">money.</span> </p>
<p><span m="775790">PROFESSOR: Oh,</span> <span m="776100">very</span> <span m="776400">good,</span> <span m="776620">that's</span> <span m="776850">good.</span> <span m="777120">That's</span> <span m="777290">what</span> <span m="777390">we</span> <span m="777450">like</span> <span m="777610">to</span> <span m="777690">hear</span> <span m="777890">here.</span> </p>
<p><span m="780030">AUDIENCE: That's what he</span> <span m="780450">likes to hear.</span> </p>
<p><span m="781600">PROFESSOR: Everybody</span> <span m="782040">make</span> <span m="782250">a</span> <span m="782300">choice.</span> <span m="783590">Heads</span> <span m="783900">or</span> <span m="783960">tails?</span> <span m="785050">Now,</span> <span m="785190">you</span> <span m="785300">all</span> <span m="785430">want</span> <span m="785550">to</span> <span m="785610">be</span> <span m="785690">thinking,</span> <span m="786240">OK,</span> <span m="786690">what's</span> <span m="786880">going</span> <span m="787120">on</span> <span m="787340">here?</span> <span m="789660">We're going</span> <span m="789780">to</span> <span m="790060">figure</span> <span m="790240">out</span> <span m="790320">expectations</span> <span m="790930">in</span> <span m="791000">a</span> <span m="791040">minute,</span> <span m="791400">do</span> <span m="791500">the</span> <span m="791580">tree</span> <span m="791790">method.</span> <span m="792110">Is</span> <span m="792210">there</span> <span m="792330">any</span> <span m="792900">catch</span> <span m="793230">going</span> <span m="793480">on?</span> <span m="795110">Not</span> <span m="795330">that</span> <span m="795430">we</span> <span m="795530">would</span> <span m="795840">have</span> <span m="796070">a</span> <span m="796110">catch</span> <span m="796430">here.</span> </p>
<p><span m="796580">OK,</span> <span m="796820">put</span> <span m="796990">a</span> <span m="797070">heads</span> <span m="797270">or</span> <span m="797320">tails</span> <span m="797670">down.</span> <span m="799150">OK,</span> <span m="799590">coin</span> <span m="799870">toss</span> <span m="800170">please.</span> </p>
<p><span m="803185">AUDIENCE: Tails.</span> </p>
<p><span m="804210">PROFESSOR: Tails.</span> <span m="804920">What</span> <span m="805030">do</span> <span m="805090">we</span> <span m="805190">got?</span> <span m="806160">A</span> <span m="806290">winner,</span> <span m="806860">a</span> <span m="806940">winner,</span> <span m="808480">and</span> <span m="808620">a</span> <span m="808680">loser.</span> <span m="811150">So we</span> <span m="811644">got</span> <span m="812140">tails,</span> <span m="814360">tails,</span> <span m="814770">tails.</span> <span m="815130">Nick</span> <span m="815470">has</span> <span m="816760">lost</span> <span m="817130">another</span> <span m="817670">$2.</span> <span m="820210">Martyna's</span> <span m="820830">cruising</span> <span m="821880">and</span> <span m="822040">Adam</span> <span m="822330">is</span> <span m="822450">back</span> <span m="822780">to</span> <span m="822860">even.</span> <span m="823840">Good</span> <span m="824240">job.</span> </p>
<p><span m="826900">AUDIENCE: Can I  cash</span> <span m="827303">out now?</span> </p>
<p><span m="828110">PROFESSOR: Well,</span> <span m="828340">you're</span> <span m="828550">only</span> <span m="828790">even.</span> <span m="829040">You've</span> <span m="829140">got</span> <span m="829230">to</span> <span m="829290">get</span> <span m="829420">ahead</span> <span m="829830">here.</span> <span m="830430">All</span> <span m="830590">right,</span> <span m="830810">let's</span> <span m="830950">do</span> <span m="831090">it</span> <span m="831150">again.</span> <span m="837790">$2.</span> <span m="838330">Everybody</span> <span m="838580">got</span> <span m="838730">$2</span> <span m="839110">on</span> <span m="839170">the</span> <span m="839230">table?</span> <span m="842650">I'm</span> <span m="842730">going</span> <span m="842850">to have to</span> <span m="843080">loan.</span> <span m="844120">You</span> <span m="844730">got</span> <span m="844830">money?</span> <span m="845460">You</span> <span m="845570">got</span> <span m="845770">it.</span> <span m="845870">OK.</span> <span m="848100">See,</span> <span m="848270">my</span> <span m="848390">hope</span> <span m="848600">is</span> <span m="848740">Nick</span> <span m="848930">loses</span> <span m="849230">track.</span> <span m="851150">OK,</span> <span m="852030">make</span> <span m="852240">a</span> <span m="852290">choice</span> <span m="852640">please.</span> <span m="859760">Coin</span> <span m="860050">toss,</span> <span m="860400">please.</span> </p>
<p><span m="862690">AUDIENCE: Tails.</span> </p>
<p><span m="863520">PROFESSOR: Tails.</span> <span m="864000">What</span> <span m="864110">do</span> <span m="864170">we</span> <span m="864250">got?</span> <span m="865770">Heads.</span> <span m="866660">Martyna,</span> <span m="867070">Nick!</span> <span m="868746">Oh,</span> <span m="869170">for</span> <span m="869360">goodness</span> <span m="869850">sake.</span> </p>
<p><span m="871360">AUDIENCE: She's psychic,</span> <span m="871855">I tell</span> <span m="872350">you.</span> </p>
<p><span m="874330">PROFESSOR: Martyna</span> <span m="874990">wins</span> <span m="875370">again.</span> <span m="876580">You</span> <span m="876700">didn't</span> <span m="876840">make</span> <span m="877030">a</span> <span m="877090">deal</span> <span m="877360">with</span> <span m="877490">him,</span> <span m="877988">did you?</span> </p>
<p><span m="880460">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="882820">PROFESSOR: Nick</span> <span m="883190">and</span> <span m="883300">Martyna</span> <span m="883740">are</span> <span m="883980">perfect</span> <span m="885750">and</span> <span m="886120">Adam</span> <span m="886460">is</span> <span m="886570">down</span> <span m="886850">again.</span> <span m="890670">Let's</span> <span m="891310">collect</span> <span m="891630">your</span> <span m="891730">money.</span> <span m="893050">She</span> <span m="893250">gets</span> <span m="893440">it</span> <span m="893520">all.</span> <span m="894440">Oh,</span> <span m="894680">she</span> <span m="894830">got</span> <span m="895550">four</span> <span m="896120">here.</span> <span m="896380">Yeah,</span> <span m="896580">she</span> <span m="896900">made</span> <span m="897100">four.</span> <span m="897590">Good</span> <span m="897840">point.</span> <span m="898590">Because</span> <span m="898790">she</span> <span m="898910">won it</span> <span m="899210">all.</span> <span m="900410">Very</span> <span m="900690">good.</span> <span m="901610">All</span> <span m="901940">right,</span> <span m="902220">let's</span> <span m="902500">put</span> <span m="902770">the</span> <span m="902850">next</span> <span m="903120">wager</span> <span m="903490">up.</span> <span m="907970">Nick,</span> <span m="908200">if</span> <span m="908290">I</span> <span m="908420">were</span> <span m="908510">you,</span> <span m="908770">I</span> <span m="908870">wouldn't</span> <span m="908980">do</span> <span m="909050">much</span> <span m="909280">gambling</span> <span m="909660">in</span> <span m="909770">life.</span> <span m="910670">[LAUGHS]</span> <span m="912140">You</span> <span m="912240">got</span> <span m="912380">any</span> <span m="912490">money</span> <span m="912720">left</span> <span m="912970">Adam?</span> </p>
<p><span m="913970">AUDIENCE: I'm going</span> <span m="914470">to have a</span> <span m="914970">big win.</span> </p>
<p><span m="915810">PROFESSOR: There</span> <span m="916070">it is.</span> <span m="916540">Working</span> <span m="916750">on</span> <span m="916880">the</span> <span m="916950">big</span> <span m="917170">win.</span> <span m="917470">There</span> <span m="917650">we</span> <span m="917770">go.</span> </p>
<p><span m="918660">AUDIENCE: I'm running</span> <span m="919135">out of</span> <span m="919610">ones</span> <span m="920085">here.</span> </p>
<p><span m="921510">PROFESSOR: Martyna</span> <span m="922040">can</span> <span m="922190">make</span> <span m="923400">change</span> <span m="923810">for</span> <span m="924030">you.</span> </p>
<p><span m="924473">AUDIENCE: Thanks,</span> <span m="924916">Martyna.</span> <span m="927950">Do you</span> <span m="928410">have change</span> <span m="928870">for a</span> <span m="929330">10?</span> </p>
<p><span m="932010">PROFESSOR: All</span> <span m="932290">right.</span> <span m="934520">There</span> <span m="934750">we</span> <span m="934860">go.</span> <span m="936680">A</span> <span m="936710">couple</span> <span m="936970">bucks</span> <span m="937240">there.</span> <span m="937530">And</span> <span m="937620">then</span> <span m="937810">make</span> <span m="938050">your</span> <span m="938180">selections.</span> <span m="942260">Okey doke.</span> <span m="943380">It's a</span> <span m="943640">pretty</span> <span m="943820">fair</span> <span m="944080">game.</span> <span m="944550">They</span> <span m="944630">put</span> <span m="944820">the</span> <span m="944880">money</span> <span m="945110">in,</span> <span m="945190">the</span> <span m="945260">winners</span> <span m="945550">split</span> <span m="945760">the</span> <span m="945840">pot.</span> <span m="946860">You</span> <span m="947010">guys</span> <span m="947310">flip</span> <span m="947510">the</span> <span m="947590">coin</span> <span m="948190">after</span> <span m="948540">they</span> <span m="948650">make</span> <span m="948860">their</span> <span m="948970">selections.</span> <span m="950760">OK,</span> <span m="951160">make</span> <span m="951370">a</span> <span m="951420">selection.</span> <span m="952360">Martyna</span> <span m="952760">has</span> <span m="953010">hers.</span> <span m="954860">Adam's</span> <span m="955240">ready</span> <span m="955820">and</span> <span m="956200">Nick</span> <span m="956410">is</span> <span m="956550">ready.</span> <span m="956820">Can we</span> <span m="956940">have</span> <span m="957030">a</span> <span m="957070">coin</span> <span m="957360">toss?</span> </p>
<p><span m="959550">AUDIENCE: Tails.</span> </p>
<p><span m="960550">PROFESSOR: Tails.</span> <span m="961390">What</span> <span m="961490">do</span> <span m="961550">we</span> <span m="961630">got?</span> <span m="963006">Nick!</span> <span m="963720">Congratulations.</span> <span m="965440">Nick</span> <span m="965680">wins</span> <span m="965960">big.</span> <span m="966450">All</span> <span m="966640">right.</span> <span m="969020">Tails,</span> <span m="970180">heads,</span> <span m="971660">heads,</span> <span m="972930">tails.</span> <span m="973790">Nick,</span> <span m="974720">$4.</span> <span m="975470">He's</span> <span m="975630">climbing</span> <span m="976070">back.</span> <span m="978050">And</span> <span m="978830">Martyna</span> <span m="979300">lost</span> <span m="979680">for</span> <span m="979740">the</span> <span m="979800">first</span> <span m="980070">time</span> <span m="980260">$2.</span> <span m="981800">Poor</span> <span m="982070">Adam</span> <span m="982380">is</span> <span m="982470">sinking</span> <span m="982860">a</span> <span m="982910">little</span> <span m="983100">bit</span> <span m="983260">here.</span> </p>
<p><span m="985320">All</span> <span m="985580">right,</span> <span m="985760">one</span> <span m="986030">more.</span> <span m="986840">So</span> <span m="987270">who</span> <span m="987440">collects?</span> <span m="987790">Nick,</span> <span m="988250">do you</span> <span m="988320">want</span> <span m="988440">to</span> <span m="988500">collect</span> <span m="988800">this?</span> <span m="989960">Start</span> <span m="990250">paying</span> <span m="990490">off</span> <span m="990640">the</span> <span m="990720">debts,</span> <span m="991040">but</span> <span m="991230">save</span> <span m="991540">$2</span> <span m="991930">for</span> <span m="992020">the</span> <span m="992090">last</span> <span m="992370">round.</span> <span m="993920">This</span> <span m="994160">yours?</span> <span m="995690">Yeah,</span> <span m="995860">there's</span> <span m="996060">a</span> <span m="996120">last</span> <span m="996390">one.</span> <span m="997280">All right,</span> <span m="997310">two</span> <span m="997460">more</span> <span m="997620">bucks.</span> <span m="997950">Last</span> <span m="998280">round.</span> <span m="1003300">OK,</span> <span m="1003710">make</span> <span m="1003940">your</span> <span m="1004060">selections.</span> <span m="1007860">Martyna's</span> <span m="1008390">ready.</span> <span m="1009920">Nick</span> <span m="1010210">is</span> <span m="1010360">ready.</span> <span m="1010700">OK,</span> <span m="1011060">Adam,</span> <span m="1011310">what</span> <span m="1011410">do</span> <span m="1011470">you</span> <span m="1011550">like?</span> <span m="1015350">Coin</span> <span m="1015650">toss.</span> </p>
<p><span m="1018290">AUDIENCE: Heads.</span> </p>
<p><span m="1019180">PROFESSOR: Heads.</span> <span m="1019670">How'd</span> <span m="1019900">you</span> <span m="1019970">do?</span> <span m="1020930">Oh</span> <span m="1021220">Adam,</span> <span m="1021560">tough</span> <span m="1021820">break.</span> <span m="1022110">Nick</span> <span m="1022440">again.</span> <span m="1023580">Nick</span> <span m="1023870">is</span> <span m="1023980">the</span> <span m="1024079">sole</span> <span m="1024480">winner.</span> <span m="1024859">So</span> <span m="1025000">he</span> <span m="1025150">collects</span> <span m="1026210">on</span> <span m="1026420">the</span> <span m="1026550">heads.</span> <span m="1031440">All</span> <span m="1031569">right,</span> <span m="1031790">so</span> <span m="1031970">it's</span> <span m="1032119">plus</span> <span m="1032520">$4</span> <span m="1033280">for</span> <span m="1033460">Nick,</span> <span m="1034640">minus</span> <span m="1035190">2</span> <span m="1035660">for</span> <span m="1035819">Martina,</span> <span m="1037119">minus</span> <span m="1037700">2</span> <span m="1039010">for</span> <span m="1039349">Adam.</span> <span m="1039880">All right,</span> <span m="1040020">let's</span> <span m="1040200">see</span> <span m="1040310">how</span> <span m="1040440">they</span> <span m="1040569">did</span> <span m="1040780">in</span> <span m="1040920">total.</span> <span m="1042980">Oh,</span> <span m="1043329">poor</span> <span m="1043640">Adam.</span> <span m="1044020">Minus</span> <span m="1044440">$6</span> <span m="1045190">here.</span> <span m="1046940">How</span> <span m="1047000">did</span> <span m="1047130">that</span> <span m="1047349">happen?</span> <span m="1047800">Martyna</span> <span m="1049510">is</span> <span m="1051220">plus</span> <span m="1051600">$6.</span> <span m="1052560">Now</span> <span m="1052780">we</span> <span m="1052870">know</span> <span m="1053000">how</span> <span m="1053180">it</span> <span m="1053250">happened.</span> <span m="1054570">And</span> <span m="1055010">Nick</span> <span m="1056140">is</span> <span m="1057390">even.</span> <span m="1060270">Even.</span> <span m="1066950">Yeah,</span> <span m="1067250">so</span> <span m="1068300">that's</span> <span m="1068640">just</span> <span m="1068870">a</span> <span m="1068910">tough</span> <span m="1069250">break</span> <span m="1069560">for</span> <span m="1069730">Adam.</span> <span m="1070050">He</span> <span m="1070110">came</span> <span m="1070400">up</span> <span m="1070550">here</span> <span m="1070870">and</span> <span m="1071120">lost</span> <span m="1071500">$6.</span> </p>
<p><span m="1073940">AUDIENCE: Interestingly,</span> <span m="1074380">there's</span> <span m="1074820">$6</span> <span m="1075260">on the table.</span> <span m="1075700">I think I</span> <span m="1076140">should take that</span> <span m="1076580">and run.</span> </p>
<p><span m="1077020">[LAUGHS]</span> </p>
<p><span m="1078020">PROFESSOR: Yeah,</span> <span m="1078240">you</span> <span m="1078380">probably</span> <span m="1078740">should.</span> <span m="1079320">But</span> <span m="1079530">I</span> <span m="1079590">need</span> <span m="1079750">to</span> <span m="1079810">get</span> <span m="1079960">paid</span> <span m="1080200">back</span> <span m="1080510">from</span> <span m="1080790">Nick</span> <span m="1081040">here.</span> <span m="1082500">Now</span> <span m="1083940">obviously</span> <span m="1084670">this</span> <span m="1084830">is</span> <span m="1084930">a</span> <span m="1084970">fair</span> <span m="1085240">game,</span> <span m="1085570">right?</span> <span m="1086420">So</span> <span m="1086640">how</span> <span m="1086800">many</span> <span m="1086970">people</span> <span m="1087270">think</span> <span m="1087480">that</span> <span m="1087630">Adam</span> <span m="1087970">was</span> <span m="1088150">just</span> <span m="1088570">unlucky?</span> <span m="1092010">A</span> <span m="1092100">few.</span> <span m="1093010">How</span> <span m="1093200">many</span> <span m="1093410">people</span> <span m="1094020">think</span> <span m="1094270">we</span> <span m="1094400">just</span> <span m="1094660">screwed</span> <span m="1095060">him</span> <span m="1095330">out</span> <span m="1095470">of</span> <span m="1095550">$6?</span> <span m="1097450">Yeah,</span> <span m="1097750">you're</span> <span m="1097980">right.</span> <span m="1099000">But</span> <span m="1099210">let's</span> <span m="1100060">do</span> <span m="1100200">the</span> <span m="1100310">analysis</span> <span m="1101570">to</span> <span m="1101690">see</span> <span m="1102120">what</span> <span m="1102330">happened</span> <span m="1102680">here.</span> <span m="1103260">Meanwhile,</span> <span m="1103540">I</span> <span m="1103620">probably</span> <span m="1104680">should</span> <span m="1104930">give</span> <span m="1105110">him</span> <span m="1105210">his</span> <span m="1105280">$6.</span> </p>
<p><span m="1106156">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="1109060">PROFESSOR: And</span> <span m="1109340">you</span> <span m="1109650">got</span> <span m="1109880">my</span> <span m="1110120">money.</span> <span m="1111376">All right,</span> <span m="1111810">so</span> <span m="1113160">I</span> <span m="1113310">have</span> <span m="1113520">a</span> <span m="1113570">gift</span> <span m="1113800">certificate,</span> <span m="1114800">if</span> <span m="1114950">you</span> <span m="1115040">would</span> <span m="1115170">like,</span> <span m="1115450">for</span> <span m="1115580">playing</span> <span m="1115940">the</span> <span m="1116010">game</span> <span m="1116330">here.</span> <span m="1116570">You</span> <span m="1116640">can</span> <span m="1116740">have</span> <span m="1116870">this</span> <span m="1117290">or</span> <span m="1117880">you</span> <span m="1117990">can</span> <span m="1118120">have</span> <span m="1118330">the</span> <span m="1118410">pocket</span> <span m="1118750">protector.</span> <span m="1120450">I</span> <span m="1120480">got</span> <span m="1120630">to</span> <span m="1120710">say</span> <span m="1120910">somebody's</span> <span m="1121260">last</span> <span m="1121580">time</span> <span m="1122250">turned</span> <span m="1122640">in</span> <span m="1122820">this</span> <span m="1123700">for</span> <span m="1123830">one</span> <span m="1124030">of</span> <span m="1124110">these.</span> <span m="1124670">The</span> <span m="1125240">nerd</span> <span m="1125570">pride</span> <span m="1125830">pocket</span> <span m="1126150">protector.</span> <span m="1126560">So</span> <span m="1126680">what</span> <span m="1126810">would</span> <span m="1126920">you</span> <span m="1127030">like</span> <span m="1127260">for</span> <span m="1127470">your</span> <span m="1128440">memento?</span> </p>
<p><span m="1130176">AUDIENCE: I'm going</span> <span m="1130624">to take</span> <span m="1131072">the</span> <span m="1131184">box</span> <span m="1131296">where</span> <span m="1131408">[? David Chena ?]</span> <span m="1131520">is</span> <span m="1131900">standing.</span> </p>
<p><span m="1133040">PROFESSOR: What?</span> </p>
<p><span m="1135820">AUDIENCE: As in</span> <span m="1136320">[INAUDIBLE].</span> </p>
<p><span m="1137280">PROFESSOR: You</span> <span m="1137380">get</span> <span m="1137570">one</span> <span m="1137740">too</span> <span m="1137940">for</span> <span m="1138080">tossing</span> <span m="1138510">the</span> <span m="1138590">coin.</span> <span m="1140110">So</span> <span m="1140320">we'll</span> <span m="1140450">pass</span> <span m="1140830">this</span> <span m="1141313">up.</span> <span m="1143990">Here</span> <span m="1144180">you go.</span> <span m="1144310">Want</span> <span m="1144450">to</span> <span m="1144530">pass</span> <span m="1144800">that</span> <span m="1144990">up</span> <span m="1145100">for</span> <span m="1145250">our</span> <span m="1145650">coin</span> <span m="1146050">tosser?</span> <span m="1147010">OK,</span> <span m="1147410">very</span> <span m="1147670">good.</span> <span m="1147840">Thanks</span> <span m="1148140">very</span> <span m="1148300">much.</span> <span m="1150440">Just</span> <span m="1150620">leave</span> <span m="1150740">the</span> <span m="1150800">money</span> <span m="1151080">here.</span> <span m="1151330">Yeah.</span> <span m="1152460">Leave</span> <span m="1152660">my</span> <span m="1152790">money</span> <span m="1153040">here.</span> <span m="1153210">Oh,</span> <span m="1153320">we</span> <span m="1153480">took</span> <span m="1153660">some</span> <span m="1153840">out of</span> <span m="1153990">your</span> <span m="1154150">wallet.</span> </p>
<p><span m="1154420">AUDIENCE: I got</span> <span m="1154690">$5 money</span> <span m="1155010">from you--</span> <span m="1155628">$5.</span> </p>
<p><span m="1156076">PROFESSOR: $5</span> <span m="1156524">from me,</span> <span m="1156972">yeah.</span> <span m="1157868">All right,</span> <span m="1158316">so</span> <span m="1158770">leave me the</span> <span m="1158840">$5.</span> </p>
<p><span m="1159312">AUDIENCE: This is your</span> <span m="1159784">$5.</span> </p>
<p><span m="1161672">PROFESSOR: Oh,</span> <span m="1162150">I</span> <span m="1162230">should</span> <span m="1162410">give them</span> <span m="1162550">back</span> <span m="1162720">their</span> <span m="1162800">$6.</span> <span m="1164570">Well,</span> <span m="1165190">otherwise</span> <span m="1165580">they</span> <span m="1165690">could</span> <span m="1165850">report</span> <span m="1166250">me,</span> <span m="1166420">and</span> <span m="1166520">I'd</span> <span m="1166670">be</span> <span m="1166890">in</span> <span m="1166980">trouble.</span> <span m="1167220">And it's</span> <span m="1167270">all</span> <span m="1167450">on</span> <span m="1167600">film.</span> <span m="1168450">That</span> <span m="1168930">would</span> <span m="1169060">not</span> <span m="1169260">be</span> <span m="1169370">good.</span> </p>
<p><span m="1171120">OK,</span> <span m="1171600">so</span> <span m="1171770">what</span> <span m="1171930">we're</span> <span m="1172010">going</span> <span m="1172130">to</span> <span m="1172210">do</span> <span m="1172340">now</span> <span m="1172700">is</span> <span m="1173290">analyze</span> <span m="1173920">the</span> <span m="1174020">game.</span> <span m="1175990">And</span> <span m="1176160">I'm</span> <span m="1176360">going</span> <span m="1176480">to</span> <span m="1176560">prove</span> <span m="1177110">to</span> <span m="1177230">you</span> <span m="1177900">he</span> <span m="1178120">was</span> <span m="1178320">just</span> <span m="1178780">unlucky,</span> <span m="1179380">despite</span> <span m="1179880">the</span> <span m="1179960">fact</span> <span m="1180320">that</span> <span m="1180420">you</span> <span m="1180520">think</span> <span m="1180670">I</span> <span m="1180740">screwed</span> <span m="1181050">him</span> <span m="1181170">here.</span> <span m="1182830">So</span> <span m="1183970">we're</span> <span m="1184110">going</span> <span m="1184230">to</span> <span m="1184300">analyze</span> <span m="1185260">Adam's</span> <span m="1185700">expected</span> <span m="1186310">winnings</span> <span m="1186730">playing</span> <span m="1187100">this</span> <span m="1187270">game.</span> <span m="1188910">So</span> <span m="1189000">let's</span> <span m="1189190">do</span> <span m="1189380">that.</span> </p>
<p><span m="1192600">So</span> <span m="1193140">we're</span> <span m="1193250">going</span> <span m="1193420">to</span> <span m="1193480">make</span> <span m="1193650">the</span> <span m="1193720">tree,</span> <span m="1194090">as</span> <span m="1194200">usual.</span> <span m="1195880">Now,</span> <span m="1195960">the</span> <span m="1196040">first</span> <span m="1196350">thing</span> <span m="1196500">we</span> <span m="1196620">have</span> <span m="1197140">is</span> <span m="1197450">Adam's</span> <span m="1197820">choice.</span> <span m="1200500">Now,</span> <span m="1200690">Adam</span> <span m="1201000">were</span> <span m="1201050">you</span> <span m="1201200">doing</span> <span m="1201540">anything</span> <span m="1202070">besides</span> <span m="1202410">sort</span> <span m="1202550">of</span> <span m="1202650">randomly</span> <span m="1203030">picking?</span> <span m="1203300">Were you</span> <span m="1203380">trying</span> <span m="1203550">to</span> <span m="1203630">psych</span> <span m="1204010">out</span> <span m="1204210">the</span> <span m="1204270">coin?</span> </p>
<p><span m="1208058">AUDIENCE: Apparently</span> <span m="1208531">I wouldn't</span> <span m="1209004">be able to</span> <span m="1209477">bribe him</span> <span m="1209950">from all the way</span> <span m="1210423">down</span> <span m="1210896">here.</span> </p>
<p><span m="1211370">PROFESSOR: Yeah,</span> <span m="1212100">and</span> <span m="1212240">you</span> <span m="1212310">did</span> <span m="1212540">lose</span> <span m="1212820">most</span> <span m="1213050">of it.</span> <span m="1213280">We'll</span> <span m="1213470">say</span> <span m="1213690">Adam</span> <span m="1213940">was</span> <span m="1214070">playing</span> <span m="1214290">50-50.</span> <span m="1216870">It's</span> <span m="1217000">a</span> <span m="1217060">reasonable</span> <span m="1217450">assumption.</span> <span m="1218850">And</span> <span m="1219100">then</span> <span m="1219260">we</span> <span m="1219450">have</span> <span m="1220900">Martyna</span> <span m="1222250">and</span> <span m="1222410">then</span> <span m="1222550">we</span> <span m="1222650">have</span> <span m="1222860">Nick</span> <span m="1223610">and</span> <span m="1223770">their</span> <span m="1223910">choices.</span> <span m="1226010">And</span> <span m="1226510">let's</span> <span m="1226690">say</span> <span m="1226890">they're</span> <span m="1227380">50-50</span> <span m="1228090">as</span> <span m="1228230">well,</span> <span m="1228560">because</span> <span m="1229070">none</span> <span m="1229330">of</span> <span m="1229400">them</span> <span m="1229530">know</span> <span m="1229720">what</span> <span m="1229860">coin</span> <span m="1230100">is</span> <span m="1230180">going</span> <span m="1230300">to</span> <span m="1230360">be</span> <span m="1230420">tossed.</span> <span m="1231280">And</span> <span m="1231440">I'm</span> <span m="1231510">not</span> <span m="1231660">going</span> <span m="1231780">to</span> <span m="1231840">draw</span> <span m="1232160">this</span> <span m="1232350">bottom</span> <span m="1232670">half</span> <span m="1232880">of</span> <span m="1232940">the</span> <span m="1233000">tree,</span> <span m="1233260">but</span> <span m="1233390">it's</span> <span m="1233550">totally</span> <span m="1233980">symmetric.</span> <span m="1235000">It</span> <span m="1235110">just</span> <span m="1235300">gets</span> <span m="1235510">too</span> <span m="1235730">deep</span> <span m="1236040">otherwise.</span> <span m="1238060">And</span> <span m="1238150">then</span> <span m="1238330">we</span> <span m="1238460">have</span> <span m="1239030">Nick's</span> <span m="1239580">choices,</span> <span m="1241200">heads,</span> <span m="1242700">tails,</span> <span m="1243550">heads,</span> <span m="1244180">tails.</span> <span m="1245230">And</span> <span m="1245390">they're</span> <span m="1245530">50-50.</span> </p>
<p><span m="1250500">And</span> <span m="1250680">then</span> <span m="1250810">we</span> <span m="1250920">have</span> <span m="1251090">the</span> <span m="1251160">coin.</span> <span m="1254070">And</span> <span m="1254190">I'll</span> <span m="1254270">start</span> <span m="1254510">drawing</span> <span m="1254790">it</span> <span m="1254880">down</span> <span m="1255180">here.</span> <span m="1255440">And</span> <span m="1255530">the</span> <span m="1255600">coin</span> <span m="1255990">can</span> <span m="1256110">be</span> <span m="1256240">heads</span> <span m="1256480">or</span> <span m="1256530">tails.</span> <span m="1257340">And</span> <span m="1257490">we're</span> <span m="1257610">going</span> <span m="1257730">to</span> <span m="1257790">assume</span> <span m="1258060">it</span> <span m="1258140">was</span> <span m="1258270">a</span> <span m="1258320">fair</span> <span m="1258690">coin</span> <span m="1259030">toss</span> <span m="1259380">back</span> <span m="1259650">there,</span> <span m="1259860">because</span> <span m="1260010">it</span> <span m="1260110">looked</span> <span m="1260260">like</span> <span m="1260410">it</span> <span m="1260470">was</span> <span m="1260610">flipping</span> <span m="1260910">a</span> <span m="1260960">bunch</span> <span m="1261180">of</span> <span m="1261260">times.</span> <span m="1262320">And</span> <span m="1262510">you</span> <span m="1262590">don't</span> <span m="1262870">look</span> <span m="1263020">like</span> <span m="1263190">Persi</span> <span m="1263470">Diaconis,</span> <span m="1264640">so</span> <span m="1265430">we're</span> <span m="1265580">going</span> <span m="1265700">to</span> <span m="1265760">assume</span> <span m="1265990">you</span> <span m="1266070">didn't</span> <span m="1266240">learn</span> <span m="1266440">how</span> <span m="1266540">to</span> <span m="1266700">flip</span> <span m="1266970">heads</span> <span m="1267260">always.</span> <span m="1269150">So</span> <span m="1269320">those</span> <span m="1269610">are</span> <span m="1269660">50-50.</span> <span m="1273790">A</span> <span m="1274150">little</span> <span m="1274450">tight</span> <span m="1274810">here.</span> <span m="1276330">Everything</span> <span m="1276840">is</span> <span m="1276970">50-50.</span> </p>
<p><span m="1283040">So</span> <span m="1283480">the</span> <span m="1283580">probabilities</span> <span m="1284420">are</span> <span m="1284530">all</span> <span m="1286160">1/16.</span> <span m="1287680">I</span> <span m="1287780">got</span> <span m="1287990">16</span> <span m="1288790">possible</span> <span m="1289320">outcomes.</span> <span m="1290670">2</span> <span m="1291030">by</span> <span m="1291180">2</span> <span m="1291430">by</span> <span m="1291570">2</span> <span m="1291800">by</span> <span m="1291960">2.</span> <span m="1292330">They're</span> <span m="1292470">all</span> <span m="1292720">equally</span> <span m="1293030">likely.</span> <span m="1294200">So</span> <span m="1294240">all</span> <span m="1294540">the</span> <span m="1294610">probabilities</span> <span m="1295300">are</span> <span m="1295370">1/16.</span> </p>
<p><span m="1301970">And</span> <span m="1302060">now</span> <span m="1302270">let's</span> <span m="1302490">see</span> <span m="1302850">the</span> <span m="1304070">winnings.</span> <span m="1305380">So</span> <span m="1305560">we'll</span> <span m="1305720">take</span> <span m="1305990">Adam's</span> <span m="1306460">game.</span> <span m="1309170">It'll</span> <span m="1309290">turn</span> <span m="1309540">out</span> <span m="1309610">to</span> <span m="1309670">be</span> <span m="1309760">negative.</span> <span m="1313790">Now,</span> <span m="1314000">in</span> <span m="1314080">this</span> <span m="1314280">case</span> <span m="1314610">if</span> <span m="1314740">Adam's</span> <span m="1315200">heads</span> <span m="1316470">and</span> <span m="1317590">Martyna</span> <span m="1317875">and</span> <span m="1318160">Nick</span> <span m="1318440">are</span> <span m="1318570">heads</span> <span m="1318880">and</span> <span m="1318990">the</span> <span m="1319050">coin</span> <span m="1319400">comes</span> <span m="1319620">up</span> <span m="1319720">heads,</span> <span m="1321980">they</span> <span m="1322160">all</span> <span m="1322400">win,</span> <span m="1322990">they</span> <span m="1323130">split</span> <span m="1323400">the</span> <span m="1323480">pot.</span> <span m="1323880">So</span> <span m="1324010">how</span> <span m="1324200">much</span> <span m="1324500">is</span> <span m="1324630">the</span> <span m="1324730">gain</span> <span m="1325200">for</span> <span m="1325350">Adam</span> <span m="1325710">in</span> <span m="1325790">that</span> <span m="1325960">case?</span> <span m="1327670">0.</span> <span m="1330630">Now</span> <span m="1331390">they</span> <span m="1331530">all</span> <span m="1331800">guessed</span> <span m="1332150">heads,</span> <span m="1332580">but</span> <span m="1332700">the</span> <span m="1332770">coin</span> <span m="1333100">comes</span> <span m="1333310">out</span> <span m="1333520">tails,</span> <span m="1334600">so</span> <span m="1334760">they</span> <span m="1334860">split</span> <span m="1335160">the</span> <span m="1335250">pot.</span> <span m="1335720">What's</span> <span m="1335960">Adam's</span> <span m="1336290">gain</span> <span m="1336560">there?</span> <span m="1339150">0,</span> <span m="1339810">because</span> <span m="1339980">they</span> <span m="1340380">just</span> <span m="1340530">split</span> <span m="1340710">the</span> <span m="1340790">pot</span> <span m="1341030">again.</span> <span m="1341240">He</span> <span m="1341330">gets</span> <span m="1341490">his</span> <span m="1341630">$2</span> <span m="1342090">back.</span> <span m="1342350">That's</span> <span m="1342520">0.</span> </p>
<p><span m="1345010">Now</span> <span m="1345340">we</span> <span m="1345470">have</span> <span m="1345800">a</span> <span m="1345850">case</span> <span m="1346260">where</span> <span m="1346610">Nick</span> <span m="1347700">is</span> <span m="1347900">tails,</span> <span m="1349430">Adam</span> <span m="1350340">and</span> <span m="1350470">Martyna</span> <span m="1350890">were</span> <span m="1351010">heads,</span> <span m="1351980">and</span> <span m="1352110">it</span> <span m="1352200">comes</span> <span m="1352510">out</span> <span m="1352720">heads.</span> <span m="1353100">So</span> <span m="1353280">Nick</span> <span m="1353600">is</span> <span m="1353820">losing</span> <span m="1354230">here.</span> <span m="1354870">The</span> <span m="1354990">pot</span> <span m="1355410">is</span> <span m="1355520">split</span> <span m="1355860">by</span> <span m="1356030">Adam and</span> <span m="1356440">Martyna.</span> <span m="1357500">What</span> <span m="1357670">does</span> <span m="1357800">Adam</span> <span m="1358100">get</span> <span m="1358380">as</span> <span m="1358490">a</span> <span m="1358550">profit?</span> <span m="1360570">1.</span> <span m="1361040">He</span> <span m="1361160">gets</span> <span m="1361370">half</span> <span m="1361650">the</span> <span m="1361750">part</span> <span m="1362200">because</span> <span m="1362350">he</span> <span m="1362420">splits</span> <span m="1362720">with</span> <span m="1362820">Martyna.</span> <span m="1363290">That's</span> <span m="1363510">$3</span> <span m="1364350">he</span> <span m="1364470">put</span> <span m="1364670">into,</span> <span m="1365520">he</span> <span m="1365630">gets</span> <span m="1365870">1,</span> <span m="1366660">plus</span> <span m="1366970">1.</span> </p>
<p><span m="1369080">Now</span> <span m="1369470">it</span> <span m="1369540">turns</span> <span m="1369860">out</span> <span m="1370350">same</span> <span m="1370700">scenario</span> <span m="1371190">of</span> <span m="1371280">guesses,</span> <span m="1371710">but</span> <span m="1371820">it</span> <span m="1371880">comes</span> <span m="1372110">out</span> <span m="1372300">tails,</span> <span m="1372840">so</span> <span m="1372960">Nick</span> <span m="1373330">wins</span> <span m="1373590">everything.</span> <span m="1374740">What's</span> <span m="1375020">Adam's</span> <span m="1375370">status</span> <span m="1375780">here?</span> <span m="1377550">Minus</span> <span m="1378130">2.</span> <span m="1378360">He</span> <span m="1378470">bet</span> <span m="1378770">$2,</span> <span m="1379250">he</span> <span m="1379360">lost</span> <span m="1379720">it.</span> <span m="1381800">Then</span> <span m="1382010">we</span> <span m="1382160">have</span> <span m="1383430">heads</span> <span m="1383840">for</span> <span m="1384040">Adam,</span> <span m="1385900">tails</span> <span m="1386410">for</span> <span m="1386510">Martyna,</span> <span m="1387000">heads</span> <span m="1387320">for</span> <span m="1387420">Nick.</span> <span m="1388330">Coin</span> <span m="1388670">comes</span> <span m="1388900">up</span> <span m="1389070">heads,</span> <span m="1389990">Adam</span> <span m="1390380">splits</span> <span m="1390750">with</span> <span m="1390900">Nick,</span> <span m="1392580">he</span> <span m="1392780">gets</span> <span m="1393080">a</span> <span m="1393120">net</span> <span m="1393440">of</span> <span m="1393530">$1.</span> <span m="1395550">Same</span> <span m="1395900">scenario</span> <span m="1396370">guesses,</span> <span m="1396830">but</span> <span m="1396950">now</span> <span m="1397150">it's</span> <span m="1397300">tails,</span> <span m="1397595">so</span> <span m="1397890">Martina</span> <span m="1398410">wins</span> <span m="1398740">everything,</span> <span m="1399940">Adam</span> <span m="1400300">loses</span> <span m="1400640">$2.</span> </p>
<p><span m="1402770">Now</span> <span m="1402980">we</span> <span m="1403070">go</span> <span m="1403180">down</span> <span m="1403390">to</span> <span m="1403460">here</span> <span m="1403800">where</span> <span m="1404030">it's</span> <span m="1404180">heads</span> <span m="1404470">for</span> <span m="1404620">Adam,</span> <span m="1406110">tails</span> <span m="1406690">for</span> <span m="1406780">Martyna</span> <span m="1407280">and</span> <span m="1407420">Nick,</span> <span m="1408650">and</span> <span m="1408970">heads</span> <span m="1409550">comes</span> <span m="1409840">up,</span> <span m="1410090">so</span> <span m="1410340">Adam</span> <span m="1410700">wins</span> <span m="1410930">the</span> <span m="1411010">whole</span> <span m="1411240">thing.</span> <span m="1411520">What</span> <span m="1411650">does</span> <span m="1411750">he</span> <span m="1411830">get</span> <span m="1411980">in</span> <span m="1412070">this</span> <span m="1412210">case?</span> <span m="1414430">Plus</span> <span m="1414740">4.</span> <span m="1415020">He</span> <span m="1415090">wins</span> <span m="1415320">the</span> <span m="1415390">whole</span> <span m="1415590">part</span> <span m="1415810">of</span> <span m="1415890">6</span> <span m="1416210">minus</span> <span m="1416490">2.</span> <span m="1416640">So</span> <span m="1416750">he</span> <span m="1416810">gets</span> <span m="1416990">4</span> <span m="1417350">net.</span> <span m="1418410">And</span> <span m="1418560">then</span> <span m="1418720">lastly,</span> <span m="1420020">in</span> <span m="1420160">this</span> <span m="1420310">scenario it</span> <span m="1420810">comes</span> <span m="1421030">up</span> <span m="1421190">tails</span> <span m="1422690">and</span> <span m="1422900">Martina</span> <span m="1423185">and</span> <span m="1423470">Nick</span> <span m="1423820">split</span> <span m="1424070">the</span> <span m="1424150">pot,</span> <span m="1425130">Adam</span> <span m="1425530">loses.</span> <span m="1426920">Minus</span> <span m="1427300">2.</span> <span m="1428810">And</span> <span m="1428940">then</span> <span m="1429050">the</span> <span m="1429120">same</span> <span m="1429570">thing</span> <span m="1429820">is</span> <span m="1429940">happening</span> <span m="1430360">down</span> <span m="1430630">here.</span> <span m="1432030">Same</span> <span m="1432620">thing.</span> <span m="1432860">It's just</span> <span m="1433110">everything</span> <span m="1433480">is</span> <span m="1434230">reversed</span> <span m="1434780">by</span> <span m="1434910">symmetry.</span> </p>
<p><span m="1436800">Now,</span> <span m="1437290">they're</span> <span m="1437460">all</span> <span m="1437740">equally</span> <span m="1438050">likely.</span> <span m="1439290">So</span> <span m="1439400">what</span> <span m="1439580">we</span> <span m="1439690">do</span> <span m="1439880">to</span> <span m="1439940">compute</span> <span m="1440310">the</span> <span m="1440400">expected</span> <span m="1441010">gain</span> <span m="1441380">is</span> <span m="1441500">take</span> <span m="1441780">each</span> <span m="1442020">value</span> <span m="1442800">times</span> <span m="1443110">its</span> <span m="1443220">probability</span> <span m="1444340">and</span> <span m="1444510">add</span> <span m="1444610">them</span> <span m="1444790">up.</span> <span m="1445290">So 0</span> <span m="1445620">times</span> <span m="1445880">1/16</span> <span m="1446315">plus</span> <span m="1446660">0</span> <span m="1446970">times</span> <span m="1447190">1/16</span> <span m="1448420">plus</span> <span m="1448610">1 times</span> <span m="1448970">1/16</span> <span m="1449690">minus</span> <span m="1450000">2</span> <span m="1450150">times</span> <span m="1450390">1/16</span> <span m="1450880">and</span> <span m="1451000">so</span> <span m="1451190">forth.</span> <span m="1452190">So</span> <span m="1452230">it's</span> <span m="1452340">easier</span> <span m="1452610">just</span> <span m="1452790">to</span> <span m="1452860">add</span> <span m="1453120">these</span> <span m="1453330">up</span> <span m="1453510">and</span> <span m="1453600">multiply</span> <span m="1454580">by</span> <span m="1454740">1/16.</span> <span m="1455450">And</span> <span m="1455630">when we</span> <span m="1455730">do,</span> <span m="1456050">we</span> <span m="1456180">get</span> <span m="1458000">0.</span> <span m="1458430">We</span> <span m="1458530">get</span> <span m="1458660">0.</span> <span m="1460070">If</span> <span m="1460140">I</span> <span m="1460260">add</span> <span m="1460390">all</span> <span m="1460480">these</span> <span m="1460650">up, I</span> <span m="1460830">get</span> <span m="1461000">0.</span> <span m="1461510">The</span> <span m="1461590">same</span> <span m="1461860">thing</span> <span m="1462040">for</span> <span m="1462140">down</span> <span m="1462400">here.</span> <span m="1463420">So</span> <span m="1463610">the</span> <span m="1463740">expected</span> <span m="1465600">gain</span> <span m="1467740">for</span> <span m="1468000">Adam</span> <span m="1471180">is</span> <span m="1471260">0.</span> <span m="1474050">And</span> <span m="1474150">that</span> <span m="1474420">is</span> <span m="1474730">a</span> <span m="1474860">fair</span> <span m="1475380">game.</span> </p>
<p><span m="1479320">What</span> <span m="1479480">do</span> <span m="1479540">you</span> <span m="1479630">think?</span> <span m="1480822">Do</span> <span m="1481250">you</span> <span m="1481370">think</span> <span m="1481560">we'd</span> <span m="1481650">go</span> <span m="1481790">to</span> <span m="1481860">all</span> <span m="1482000">that</span> <span m="1482150">trouble</span> <span m="1482430">just</span> <span m="1482620">to</span> <span m="1482700">play</span> <span m="1482920">a</span> <span m="1482970">fair</span> <span m="1483200">game?</span> <span m="1483670">Or</span> <span m="1483960">do you</span> <span m="1484030">think</span> <span m="1484250">we're</span> <span m="1484360">trying</span> <span m="1484620">to</span> <span m="1484710">set</span> <span m="1484970">him</span> <span m="1485090">up</span> <span m="1485240">and</span> <span m="1485350">take</span> <span m="1485560">his</span> <span m="1485690">money?</span> <span m="1486800">Yeah.</span> </p>
<p><span m="1488666">AUDIENCE: Martyna</span> <span m="1489020">and</span> <span m="1489260">Nick</span> <span m="1489660">are</span> <span m="1489750">alternating,</span> <span m="1491890">so the</span> <span m="1491990">branch</span> <span m="1492270">with</span> <span m="1492610">the</span> <span m="1492680">plus</span> <span m="1492960">4</span> <span m="1493270">goes</span> <span m="1493530">away.</span> </p>
<p><span m="1494360">PROFESSOR: Oh,</span> <span m="1494500">that's</span> <span m="1494750">interesting.</span> <span m="1495240">Look</span> <span m="1495390">what</span> <span m="1495530">happened</span> <span m="1495940">here.</span> <span m="1499588">Nick</span> <span m="1500500">and</span> <span m="1500680">Martina,</span> <span m="1502820">well,</span> <span m="1507446">they're</span> <span m="1507920">opposite.</span> <span m="1509350">That's</span> <span m="1509650">opposite.</span> <span m="1511290">That's</span> <span m="1511610">opposite.</span> <span m="1512090">That's</span> <span m="1512380">opposite.</span> <span m="1514180">You</span> <span m="1514290">don't</span> <span m="1514460">suppose</span> <span m="1514760">they</span> <span m="1514900">planned</span> <span m="1515310">that,</span> <span m="1515540">do</span> <span m="1515680">you?</span> <span m="1519610">And</span> <span m="1519760">how</span> <span m="1519920">could</span> <span m="1520060">that</span> <span m="1520210">possibly</span> <span m="1520710">help</span> <span m="1521990">that</span> <span m="1522180">they</span> <span m="1522330">just</span> <span m="1522620">happened</span> <span m="1523030">to</span> <span m="1523110">guess</span> <span m="1523490">opposite?</span> </p>
<p><span m="1524426">AUDIENCE: One of them always</span> <span m="1524894">wins.</span> </p>
<p><span m="1525830">PROFESSOR: One</span> <span m="1526100">of</span> <span m="1526180">them</span> <span m="1526300">always</span> <span m="1526670">wins.</span> <span m="1527920">And</span> <span m="1528130">what</span> <span m="1528230">does</span> <span m="1528340">that</span> <span m="1528520">mean</span> <span m="1528740">for</span> <span m="1528850">poor</span> <span m="1529130">Adam?</span> </p>
<p><span m="1530357">AUDIENCE: He never takes the</span> <span m="1530764">whole</span> <span m="1531171">pot.</span> </p>
<p><span m="1531580">PROFESSOR: He</span> <span m="1531760">never</span> <span m="1532240">takes</span> <span m="1532590">the</span> <span m="1532680">whole</span> <span m="1532950">pot.</span> <span m="1534660">Well,</span> <span m="1535090">all</span> <span m="1535260">right.</span> <span m="1536430">Let's</span> <span m="1536610">see</span> <span m="1536830">if</span> <span m="1536940">that</span> <span m="1537120">changed</span> <span m="1537560">anything</span> <span m="1537910">here.</span> <span m="1540050">So</span> <span m="1540950">if</span> <span m="1541470">Nick</span> <span m="1541770">and</span> <span m="1541860">Martyna</span> <span m="1543420">are</span> <span m="1543600">always</span> <span m="1543950">opposite,</span> <span m="1544450">that</span> <span m="1544630">means</span> <span m="1545420">some</span> <span m="1545710">of</span> <span m="1545780">these</span> <span m="1545960">branches</span> <span m="1546420">can't</span> <span m="1546740">occur.</span> <span m="1548260">So if</span> <span m="1548470">Martyna's</span> <span m="1549100">heads,</span> <span m="1549770">Nick</span> <span m="1550010">is</span> <span m="1550460">tails,</span> <span m="1551020">this</span> <span m="1551240">is</span> <span m="1551380">out.</span> </p>
<p><span m="1551760">So</span> <span m="1551860">this</span> <span m="1552030">is</span> <span m="1552140">happening</span> <span m="1552490">with</span> <span m="1552600">probability</span> <span m="1553040">1</span> <span m="1553360">now.</span> <span m="1555100">So</span> <span m="1555410">these</span> <span m="1555760">points</span> <span m="1557740">are</span> <span m="1558360">at</span> <span m="1558510">1/8</span> <span m="1559050">each</span> <span m="1561090">instead</span> <span m="1561430">of</span> <span m="1561500">1/16.</span> <span m="1562210">And</span> <span m="1562310">these</span> <span m="1562560">go</span> <span m="1562700">to</span> <span m="1562780">0</span> <span m="1564140">because</span> <span m="1564790">that</span> <span m="1565010">branch</span> <span m="1565320">can't</span> <span m="1565540">happen.</span> <span m="1566770">Same</span> <span m="1567060">thing</span> <span m="1567260">here.</span> <span m="1567490">When</span> <span m="1567620">you</span> <span m="1567700">go</span> <span m="1567910">tails</span> <span m="1568185">to</span> <span m="1568460">Martyna,</span> <span m="1569470">Nick</span> <span m="1569840">can't</span> <span m="1570150">be</span> <span m="1570270">tails,</span> <span m="1571290">has</span> <span m="1571540">to</span> <span m="1571630">be</span> <span m="1571770">heads.</span> <span m="1573110">So</span> <span m="1573300">these</span> <span m="1573620">go</span> <span m="1573760">to</span> <span m="1573860">1/8.</span> <span m="1576590">These</span> <span m="1577000">go</span> <span m="1577190">to</span> <span m="1577280">0.</span> </p>
<p><span m="1581180">Now,</span> <span m="1581410">you</span> <span m="1581530">notice</span> <span m="1582340">this</span> <span m="1582550">isn't</span> <span m="1582770">working</span> <span m="1583120">out</span> <span m="1583290">so</span> <span m="1583420">well</span> <span m="1583630">for</span> <span m="1583790">Adam</span> <span m="1584190">because</span> <span m="1585410">we're</span> <span m="1585540">putting</span> <span m="1585900">more</span> <span m="1586130">weight</span> <span m="1586480">here</span> <span m="1586830">where</span> <span m="1587000">he's</span> <span m="1587210">got</span> <span m="1587390">a</span> <span m="1587460">net</span> <span m="1587750">negative</span> <span m="1588640">compared</span> <span m="1589100">to</span> <span m="1589220">an</span> <span m="1589610">even</span> <span m="1590510">situation.</span> <span m="1591750">And</span> <span m="1591930">here</span> <span m="1592210">he's</span> <span m="1593460">gone</span> <span m="1593780">from</span> <span m="1593910">a</span> <span m="1593970">positive</span> <span m="1594520">situation,</span> <span m="1595030">that's</span> <span m="1595200">getting</span> <span m="1595420">wiped</span> <span m="1595750">out</span> <span m="1596010">to</span> <span m="1596110">a</span> <span m="1596150">net</span> <span m="1596370">negative.</span> </p>
<p><span m="1597380">So</span> <span m="1597430">let's</span> <span m="1597640">compute</span> <span m="1597980">his</span> <span m="1598100">probability</span> <span m="1598700">now.</span> <span m="1599960">And</span> <span m="1600070">I'll</span> <span m="1600130">get</span> <span m="1600250">the</span> <span m="1600310">same</span> <span m="1600530">contribution</span> <span m="1601100">down</span> <span m="1601360">here.</span> <span m="1602480">I've</span> <span m="1602680">got</span> <span m="1603320">1</span> <span m="1603650">minus</span> <span m="1604010">2</span> <span m="1604340">plus</span> <span m="1604580">1</span> <span m="1604760">of</span> <span m="1604820">0</span> <span m="1605090">minus</span> <span m="1605450">2</span> <span m="1605820">is</span> <span m="1606850">negative</span> <span m="1608300">2.</span> <span m="1609760">I'll</span> <span m="1609910">get</span> <span m="1610060">a</span> <span m="1610090">negative</span> <span m="1610500">2</span> <span m="1610690">down</span> <span m="1610930">here</span> <span m="1611080">times</span> <span m="1611350">1/8.</span> <span m="1612760">In</span> <span m="1612900">this</span> <span m="1613090">case,</span> <span m="1613390">I'll</span> <span m="1613480">have</span> <span m="1613730">the</span> <span m="1613930">expected</span> <span m="1614630">gain</span> <span m="1615080">for</span> <span m="1615440">Adam</span> <span m="1617060">is</span> <span m="1617280">going</span> <span m="1617570">to</span> <span m="1617690">be</span> <span m="1619720">minus.</span> <span m="1620330">Let's</span> <span m="1620510">make</span> <span m="1620670">sure</span> <span m="1620810">I</span> <span m="1620850">do</span> <span m="1620990">this</span> <span m="1621220">right.</span> <span m="1622270">It's</span> <span m="1622480">going</span> <span m="1622610">to</span> <span m="1622700">be</span> <span m="1624280">2</span> <span m="1624650">for</span> <span m="1624770">the</span> <span m="1624850">top</span> <span m="1625100">and</span> <span m="1625210">bottom</span> <span m="1626180">times</span> <span m="1626630">1/8</span> <span m="1629130">times</span> <span m="1630630">these</span> <span m="1630940">guys.</span> <span m="1632560">1</span> <span m="1632890">minus</span> <span m="1633230">2</span> <span m="1633570">plus</span> <span m="1633930">1</span> <span m="1634900">minus</span> <span m="1635330">2.</span> <span m="1637110">And</span> <span m="1637270">that</span> <span m="1637500">equals</span> <span m="1639310">minus</span> <span m="1639750">2.</span> <span m="1640890">That's</span> <span m="1641050">minus</span> <span m="1641400">1/2.</span> </p>
<p><span m="1647540">So</span> <span m="1647790">in</span> <span m="1647910">fact,</span> <span m="1648800">if</span> <span m="1649040">they</span> <span m="1649220">come</span> <span m="1649430">up</span> <span m="1649620">here</span> <span m="1649850">and</span> <span m="1649970">guess</span> <span m="1650260">differently,</span> <span m="1651600">who</span> <span m="1651840">knew?</span> <span m="1653330">Now</span> <span m="1653710">the</span> <span m="1653840">expected</span> <span m="1654510">gain</span> <span m="1654800">for</span> <span m="1654950">Adam</span> <span m="1655310">is</span> <span m="1655420">minus</span> <span m="1655870">$0.50.</span> <span m="1656610">Every</span> <span m="1656900">time</span> <span m="1657150">he</span> <span m="1657280">plays,</span> <span m="1658000">he</span> <span m="1658260">expects</span> <span m="1658730">to</span> <span m="1658820">lose</span> <span m="1659320">$0.50</span> <span m="1660110">on</span> <span m="1660210">his</span> <span m="1660360">$2</span> <span m="1660840">bet.</span> <span m="1662420">That's</span> <span m="1662690">a</span> <span m="1662750">lousy</span> <span m="1663370">game</span> <span m="1664520">for</span> <span m="1664680">Adam</span> <span m="1664990">to</span> <span m="1665050">be</span> <span m="1665160">playing,</span> <span m="1666030">even</span> <span m="1666290">though</span> <span m="1666420">it</span> <span m="1666530">seemed</span> <span m="1667760">very</span> <span m="1668100">fair.</span> <span m="1670830">You</span> <span m="1670930">see</span> <span m="1671100">what's</span> <span m="1671280">going</span> <span m="1671510">on</span> <span m="1671700">here?</span> </p>
<p><span m="1672680">Now,</span> <span m="1673890">this</span> <span m="1674170">kind</span> <span m="1674410">of</span> <span m="1674490">trick</span> <span m="1675760">is</span> <span m="1675950">used</span> <span m="1676230">in</span> <span m="1676320">all</span> <span m="1676780">sorts</span> <span m="1677300">of</span> <span m="1677400">gambling</span> <span m="1677930">games.</span> <span m="1679120">Maybe</span> <span m="1679410">you've</span> <span m="1679500">probably</span> <span m="1679830">played</span> <span m="1680140">some</span> <span m="1680260">of</span> <span m="1680320">these</span> <span m="1680480">games</span> <span m="1680790">and</span> <span m="1680890">may</span> <span m="1681050">not</span> <span m="1681240">have</span> <span m="1681370">realized</span> <span m="1681800">maybe</span> <span m="1681990">somebody</span> <span m="1682390">was</span> <span m="1682550">using</span> <span m="1682850">this</span> <span m="1683370">trick</span> <span m="1683620">against</span> <span m="1684030">you.</span> </p>
<p><span m="1685240">For</span> <span m="1685260">example,</span> <span m="1685860">how</span> <span m="1686020">many</span> <span m="1686240">of</span> <span m="1686330">you</span> <span m="1686510">have</span> <span m="1686640">been</span> <span m="1687420">in</span> <span m="1687530">some</span> <span m="1687720">kind</span> <span m="1687880">of</span> <span m="1687950">sports</span> <span m="1688480">betting</span> <span m="1688910">pool?</span> <span m="1690440">It's</span> <span m="1690580">March</span> <span m="1690940">Madness,</span> <span m="1691390">you're</span> <span m="1691520">betting</span> <span m="1691840">on</span> <span m="1692020">the</span> <span m="1692770">victors</span> <span m="1693250">in</span> <span m="1693370">round</span> <span m="1693710">one.</span> <span m="1695350">It's</span> <span m="1695510">a</span> <span m="1695650">football</span> <span m="1696190">pool</span> <span m="1696480">for</span> <span m="1696580">the</span> <span m="1696660">weekend.</span> <span m="1697180">You're</span> <span m="1697290">going</span> <span m="1697440">to</span> <span m="1697500">guess</span> <span m="1697770">against</span> <span m="1698070">the</span> <span m="1698140">spread.</span> <span m="1698550">Who's</span> <span m="1698680">going</span> <span m="1698800">to</span> <span m="1698860">win?</span> <span m="1699240">All</span> <span m="1699320">right,</span> <span m="1699480">some of</span> <span m="1699570">you have</span> <span m="1699800">done</span> <span m="1700000">that.</span> <span m="1701420">All</span> <span m="1701670">right,</span> <span m="1701850">now</span> <span m="1702010">everybody</span> <span m="1702280">puts</span> <span m="1702480">$1</span> <span m="1702980">into</span> <span m="1703130">the</span> <span m="1703230">pool.</span> <span m="1704160">And</span> <span m="1704370">the</span> <span m="1704440">winner</span> <span m="1704880">is</span> <span m="1704970">the</span> <span m="1705030">guy</span> <span m="1705140">who</span> <span m="1705200">was</span> <span m="1705330">the</span> <span m="1705400">most</span> <span m="1706250">wins</span> <span m="1707110">or</span> <span m="1707320">games</span> <span m="1707690">picked</span> <span m="1707910">right.</span> <span m="1708330">All</span> <span m="1708630">the</span> <span m="1708710">winners</span> <span m="1708990">split</span> <span m="1709210">the</span> <span m="1709290">pot.</span> </p>
<p><span m="1711240">Now,</span> <span m="1711450">what</span> <span m="1711580">this</span> <span m="1711770">says,</span> <span m="1712630">by</span> <span m="1712740">doing</span> <span m="1712950">the</span> <span m="1713020">same</span> <span m="1713410">kind</span> <span m="1713630">of</span> <span m="1713700">analysis</span> <span m="1714280">we</span> <span m="1714380">just</span> <span m="1714660">did,</span> <span m="1716060">is</span> <span m="1716340">that</span> <span m="1717210">if</span> <span m="1717400">you</span> <span m="1717660">collude</span> <span m="1718320">with</span> <span m="1718490">one</span> <span m="1718740">or</span> <span m="1718830">two</span> <span m="1719090">or</span> <span m="1719130">three</span> <span m="1719430">other</span> <span m="1719600">players</span> <span m="1720010">in</span> <span m="1720070">the</span> <span m="1720130">pool</span> <span m="1720930">to</span> <span m="1721050">always</span> <span m="1721590">pick</span> <span m="1721880">differently</span> <span m="1723260">on</span> <span m="1723410">all</span> <span m="1723570">the</span> <span m="1723660">games,</span> <span m="1724730">it's</span> <span m="1724880">going</span> <span m="1725000">to</span> <span m="1725070">give</span> <span m="1725200">you</span> <span m="1725300">an</span> <span m="1725390">edge.</span> <span m="1727170">Same</span> <span m="1727580">reasoning.</span> <span m="1728330">When</span> <span m="1728500">they</span> <span m="1728660">pick</span> <span m="1728940">differently,</span> <span m="1729980">it</span> <span m="1730130">gives</span> <span m="1730410">them</span> <span m="1730760">an</span> <span m="1730870">edge</span> <span m="1731280">and</span> <span m="1731430">now</span> <span m="1731560">your</span> <span m="1731730">expected</span> <span m="1732240">return</span> <span m="1733240">is</span> <span m="1733400">bigger</span> <span m="1733620">than</span> <span m="1733750">0,</span> <span m="1734580">at</span> <span m="1734810">the</span> <span m="1734910">expense</span> <span m="1735560">of</span> <span m="1736280">the</span> <span m="1736440">other</span> <span m="1736630">players</span> <span m="1737020">who</span> <span m="1737080">just</span> <span m="1737320">go in</span> <span m="1737550">and are</span> <span m="1737660">putting</span> <span m="1737820">their</span> <span m="1737940">picks,</span> <span m="1739420">if</span> <span m="1739560">we</span> <span m="1739640">assume</span> <span m="1739960">each</span> <span m="1740120">one</span> <span m="1740310">is</span> <span m="1740430">50-50.</span> </p>
<p><span m="1742340">In</span> <span m="1742470">fact,</span> <span m="1743820">a</span> <span m="1743930">former</span> <span m="1744650">professor</span> <span m="1745120">of</span> <span m="1745180">statistics</span> <span m="1745720">here</span> <span m="1745870">at</span> <span m="1745950">MIT</span> <span m="1746590">used</span> <span m="1746890">this</span> <span m="1747230">idea,</span> <span m="1748030">a</span> <span m="1748430">guy</span> <span m="1748550">named</span> <span m="1748760">Herman</span> <span m="1749050">Chernoff,</span> <span m="1750310">in</span> <span m="1750460">the</span> <span m="1750540">1980s</span> <span m="1752030">to</span> <span m="1752160">beat</span> <span m="1753020">the</span> <span m="1753130">state</span> <span m="1753530">lottery.</span> <span m="1755190">Now,</span> <span m="1755990">everyone</span> <span m="1756400">knows</span> <span m="1756810">that</span> <span m="1756970">lotteries</span> <span m="1757600">are</span> <span m="1757660">the</span> <span m="1757740">worst</span> <span m="1758260">game</span> <span m="1758500">around</span> <span m="1758910">because</span> <span m="1759790">everybody</span> <span m="1760190">puts</span> <span m="1760430">the</span> <span m="1760510">money</span> <span m="1760880">in,</span> <span m="1761570">the</span> <span m="1761680">state</span> <span m="1762110">takes</span> <span m="1762770">half,</span> <span m="1764080">and</span> <span m="1764260">then</span> <span m="1764400">the</span> <span m="1764470">winners</span> <span m="1764960">split</span> <span m="1765390">the</span> <span m="1765460">pot.</span> <span m="1766850">So</span> <span m="1767100">it's</span> <span m="1767230">a</span> <span m="1767300">horrendous</span> <span m="1767780">game.</span> <span m="1768540">Your</span> <span m="1768740">expected</span> <span m="1769260">return</span> <span m="1769790">is</span> <span m="1769950">half</span> <span m="1770310">of</span> <span m="1770420">what</span> <span m="1770590">you</span> <span m="1770670">put</span> <span m="1770890">in.</span> <span m="1771190">So</span> <span m="1771230">you</span> <span m="1771290">expect</span> <span m="1771580">to</span> <span m="1771650">lose</span> <span m="1772000">half</span> <span m="1772220">your</span> <span m="1772340">bet,</span> <span m="1772830">because</span> <span m="1773000">the</span> <span m="1773080">state's</span> <span m="1773450">taking</span> <span m="1773850">half</span> <span m="1774730">and</span> <span m="1774860">splitting</span> <span m="1775210">the</span> <span m="1775300">remainder</span> <span m="1775870">among</span> <span m="1776560">all</span> <span m="1776810">the</span> <span m="1776890">participants.</span> </p>
<p><span m="1778830">Now,</span> <span m="1778960">what</span> <span m="1779130">Chernoff</span> <span m="1779970">realized</span> <span m="1780850">is</span> <span m="1781070">that</span> <span m="1782400">people</span> <span m="1782810">don't</span> <span m="1783300">bet</span> <span m="1783640">randomly.</span> <span m="1785170">They</span> <span m="1785360">tend</span> <span m="1785870">to</span> <span m="1785950">pick</span> <span m="1786770">the</span> <span m="1786890">same</span> <span m="1787610">sets</span> <span m="1787900">of</span> <span m="1788110">numbers.</span> <span m="1789540">It</span> <span m="1789630">might</span> <span m="1789860">be</span> <span m="1789970">a</span> <span m="1790040">birthday.</span> <span m="1790780">There's</span> <span m="1790990">only</span> <span m="1791170">so</span> <span m="1791420">many</span> <span m="1791830">birthdays</span> <span m="1792380">out</span> <span m="1792600">there.</span> <span m="1794500">Might</span> <span m="1794750">be</span> <span m="1795550">number</span> <span m="1795880">of</span> <span m="1796020">home</span> <span m="1796290">runs</span> <span m="1796540">Papi's</span> <span m="1797040">hit,</span> <span m="1797660">his</span> <span m="1797870">batting</span> <span m="1798280">average,</span> <span m="1798710">Pabelbon's</span> <span m="1799030">ERA.</span> <span m="1800530">Who</span> <span m="1801160">knows?</span> <span m="1803100">But</span> <span m="1803170">they</span> <span m="1803300">pick</span> <span m="1803580">a</span> <span m="1803670">relatively</span> <span m="1804240">small</span> <span m="1804910">set</span> <span m="1805880">of</span> <span m="1806050">numbers</span> <span m="1807160">that</span> <span m="1807300">tend</span> <span m="1807580">to</span> <span m="1808060">collapse</span> <span m="1808490">on.</span> </p>
<p><span m="1808620">In</span> <span m="1808710">fact,</span> <span m="1810780">if</span> <span m="1810870">you</span> <span m="1811140">graph</span> <span m="1812330">how</span> <span m="1812560">people</span> <span m="1812890">tend</span> <span m="1813130">to</span> <span m="1813200">pick.</span> <span m="1814370">And</span> <span m="1814520">say</span> <span m="1814700">you're</span> <span m="1814800">playing</span> <span m="1815350">pick</span> <span m="1815710">four,</span> <span m="1816010">where</span> <span m="1816100">you</span> <span m="1816170">pick</span> <span m="1816410">four</span> <span m="1816660">numbers.</span> <span m="1818190">And</span> <span m="1818330">now</span> <span m="1818510">you</span> <span m="1818670">look</span> <span m="1818890">at</span> <span m="1818950">the</span> <span m="1819020">frequency</span> <span m="1819690">with</span> <span m="1819830">which</span> <span m="1820020">the</span> <span m="1820090">numbers</span> <span m="1820450">are</span> <span m="1820510">picked.</span> <span m="1821690">Very</span> <span m="1821900">crudely</span> <span m="1822360">it</span> <span m="1822470">looks</span> <span m="1822720">something</span> <span m="1823750">like</span> <span m="1823990">this.</span> <span m="1825090">Every</span> <span m="1825360">once</span> <span m="1825570">in</span> <span m="1825630">a</span> <span m="1825660">while</span> <span m="1825760">you</span> <span m="1825850">get</span> <span m="1825970">a</span> <span m="1826030">hot</span> <span m="1826900">ticket</span> <span m="1827230">where</span> <span m="1827340">a</span> <span m="1827390">lot</span> <span m="1827630">of</span> <span m="1827710">people</span> <span m="1828490">pick</span> <span m="1828760">that</span> <span m="1828980">number.</span> </p>
<p><span m="1831890">For</span> <span m="1832020">example,</span> <span m="1832450">if</span> <span m="1832530">you're</span> <span m="1832620">picking</span> <span m="1832910">four</span> <span m="1833130">numbers,</span> <span m="1833940">MIT</span> <span m="1834440">students</span> <span m="1834850">might</span> <span m="1835130">pick</span> <span m="1836090">2,</span> <span m="1836570">4,</span> <span m="1837700">6,</span> <span m="1838170">16.</span> <span m="1839490">They</span> <span m="1839640">might</span> <span m="1839880">pick</span> <span m="1840100">that</span> <span m="1840410">and so it'd</span> <span m="1840550">be</span> <span m="1840630">a</span> <span m="1840690">big</span> <span m="1840950">spike</span> <span m="1841380">for</span> <span m="1841520">that</span> <span m="1841770">set</span> <span m="1841950">of</span> <span m="1842030">four.</span> <span m="1843070">Down</span> <span m="1843620">the</span> <span m="1843680">street</span> <span m="1844180">they're</span> <span m="1844310">probably</span> <span m="1844600">picking</span> <span m="1844840">in</span> <span m="1845060">1,</span> <span m="1845330">2,</span> <span m="1845530">3,</span> <span m="1845820">4.</span> <span m="1847430">Something</span> <span m="1847670">like</span> <span m="1847900">that</span> <span m="1848150">and there's</span> <span m="1848310">a</span> <span m="1848370">spike</span> <span m="1848760">there.</span> </p>
<p><span m="1852280">If</span> <span m="1852450">you</span> <span m="1852630">knew</span> <span m="1853170">this</span> <span m="1853370">was</span> <span m="1853520">the</span> <span m="1853630">histogram</span> <span m="1855180">of</span> <span m="1855370">what</span> <span m="1855550">people</span> <span m="1855840">were</span> <span m="1855960">picking</span> <span m="1857090">and</span> <span m="1857300">you</span> <span m="1857440">knew</span> <span m="1857790">half</span> <span m="1858110">the</span> <span m="1858220">pot</span> <span m="1858520">was</span> <span m="1858640">going</span> <span m="1858760">to</span> <span m="1858830">get</span> <span m="1858990">split</span> <span m="1859290">with</span> <span m="1859390">the</span> <span m="1859460">winners,</span> <span m="1861060">what</span> <span m="1861310">would</span> <span m="1861450">you</span> <span m="1861550">pick?</span> <span m="1863430">Would</span> <span m="1863590">you</span> <span m="1863670">pick</span> <span m="1863890">these?</span> <span m="1865440">No,</span> <span m="1865830">because</span> <span m="1865980">now</span> <span m="1866070">you're</span> <span m="1866220">splitting</span> <span m="1866520">the</span> <span m="1866590">pot</span> <span m="1866890">with</span> <span m="1866980">100</span> <span m="1867300">people.</span> <span m="1867590">That's</span> <span m="1867810">no</span> <span m="1867970">good.</span> <span m="1869190">You</span> <span m="1869340">pick</span> <span m="1869590">down</span> <span m="1869890">here.</span> <span m="1871640">And</span> <span m="1871780">now</span> <span m="1872030">when</span> <span m="1872160">you</span> <span m="1872260">win,</span> <span m="1873700">you</span> <span m="1873880">get</span> <span m="1874190">half</span> <span m="1874550">the</span> <span m="1874670">pot.</span> <span m="1875640">The</span> <span m="1875790">state</span> <span m="1876120">always</span> <span m="1876350">takes</span> <span m="1876560">half,</span> <span m="1876750">but</span> <span m="1876870">you</span> <span m="1876940">don't</span> <span m="1877060">have</span> <span m="1877160">to</span> <span m="1877230">split</span> <span m="1877640">it</span> <span m="1878080">with</span> <span m="1878280">anybody</span> <span m="1878640">else.</span> <span m="1879470">And</span> <span m="1879630">that</span> <span m="1879800">means</span> <span m="1880900">if</span> <span m="1881060">you're</span> <span m="1881170">picking</span> <span m="1881430">down</span> <span m="1881670">here,</span> <span m="1881950">your</span> <span m="1882170">expected</span> <span m="1882810">return</span> <span m="1884630">is</span> <span m="1884790">positive,</span> <span m="1885820">even</span> <span m="1886080">with</span> <span m="1886180">the</span> <span m="1886240">state</span> <span m="1886530">taking</span> <span m="1886860">half.</span> <span m="1887640">Because</span> <span m="1887820">so</span> <span m="1888140">much</span> <span m="1888360">of</span> <span m="1888420">the</span> <span m="1888480">money</span> <span m="1888830">is</span> <span m="1888970">piled</span> <span m="1889340">up</span> <span m="1889510">in</span> <span m="1889600">these</span> <span m="1889820">things.</span> </p>
<p><span m="1892330">And</span> <span m="1892430">so</span> <span m="1892700">Chernoff</span> <span m="1893160">proved</span> <span m="1893760">that</span> <span m="1894630">in</span> <span m="1895680">fact,</span> <span m="1896710">he</span> <span m="1896910">didn't</span> <span m="1897160">know</span> <span m="1897660">which</span> <span m="1898040">numbers</span> <span m="1898420">were</span> <span m="1898520">popular.</span> <span m="1899550">He</span> <span m="1899660">didn't</span> <span m="1899840">know</span> <span m="1900000">that.</span> <span m="1900960">So</span> <span m="1901150">what</span> <span m="1901290">did</span> <span m="1901390">he</span> <span m="1901470">do?</span> <span m="1902500">If</span> <span m="1902620">you</span> <span m="1902700">don't</span> <span m="1902920">know</span> <span m="1903060">where</span> <span m="1903210">the</span> <span m="1903290">spikes</span> <span m="1903720">are but</span> <span m="1903930">you're</span> <span m="1904050">trying</span> <span m="1904360">to</span> <span m="1904480">avoid</span> <span m="1904810">them</span> <span m="1904940">and</span> <span m="1905030">there's</span> <span m="1905160">not</span> <span m="1905330">very</span> <span m="1905490">many</span> <span m="1905690">spikes,</span> <span m="1906350">what</span> <span m="1906560">would</span> <span m="1906680">you</span> <span m="1906770">do</span> <span m="1907000">to</span> <span m="1907090">avoid</span> <span m="1907380">them?</span> </p>
<p><span m="1909470">Pick</span> <span m="1909800">random.</span> <span m="1911850">Because if</span> <span m="1911940">you</span> <span m="1912050">pick</span> <span m="1912260">random,</span> <span m="1912710">probably</span> <span m="1913210">you</span> <span m="1913300">miss</span> <span m="1913540">the</span> <span m="1913600">spike.</span> <span m="1913970">You're</span> <span m="1914070">down</span> <span m="1914330">here.</span> <span m="1915970">With</span> <span m="1916070">a</span> <span m="1916220">number</span> <span m="1916530">nobody</span> <span m="1916880">would</span> <span m="1917000">have</span> <span m="1917090">thought</span> <span m="1917340">to</span> <span m="1917420">pick,</span> <span m="1917680">if</span> <span m="1917750">some</span> <span m="1918020">random</span> <span m="1918400">number.</span> <span m="1919430">And</span> <span m="1919560">he</span> <span m="1919650">showed</span> <span m="1920160">that,</span> <span m="1920260">in</span> <span m="1920360">fact,</span> <span m="1920720">if</span> <span m="1920790">you</span> <span m="1920880">pick</span> <span m="1921130">randomly,</span> <span m="1922240">your</span> <span m="1922420">expected</span> <span m="1922900">gain</span> <span m="1923960">for</span> <span m="1924060">the</span> <span m="1924170">lottery</span> <span m="1924710">at</span> <span m="1924780">the</span> <span m="1924850">time</span> <span m="1926870">was</span> <span m="1927400">7%,</span> <span m="1928600">or</span> <span m="1928830">$0.7</span> <span m="1929470">on</span> <span m="1929560">the</span> <span m="1929630">dollar.</span> <span m="1930500">So</span> <span m="1930670">positive</span> <span m="1931260">even</span> <span m="1931500">with</span> <span m="1931600">the</span> <span m="1931660">state</span> <span m="1931960">taking</span> <span m="1932280">half.</span> </p>
<p><span m="1933500">Now,</span> <span m="1933740">shortly</span> <span m="1934190">after</span> <span m="1934540">that,</span> <span m="1935590">you</span> <span m="1935940">saw</span> <span m="1936280">the</span> <span m="1936410">proliferation</span> <span m="1937160">of</span> <span m="1937220">these</span> <span m="1937400">random</span> <span m="1938230">machines</span> <span m="1938800">that</span> <span m="1938890">we</span> <span m="1938970">create</span> <span m="1939340">random</span> <span m="1939660">numbers.</span> <span m="1940970">Because</span> <span m="1941140">the</span> <span m="1941220">state</span> <span m="1941530">wanted</span> <span m="1941840">to</span> <span m="1942030">balance</span> <span m="1942500">it</span> <span m="1942610">out</span> <span m="1942770">and</span> <span m="1942860">not</span> <span m="1943090">have</span> <span m="1943270">this</span> <span m="1943420">kind</span> <span m="1944730">of</span> <span m="1944800">a</span> <span m="1944850">scenario.</span> <span m="1945310">So</span> <span m="1945450">now</span> <span m="1945680">a</span> <span m="1945740">lot</span> <span m="1946020">of</span> <span m="1946080">the</span> <span m="1946150">picks</span> <span m="1946440">are</span> <span m="1946530">randomly</span> <span m="1946910">generated</span> <span m="1947940">in</span> <span m="1948070">a</span> <span m="1948120">lot</span> <span m="1948360">of</span> <span m="1948460">the</span> <span m="1949130">lotteries.</span> <span m="1954210">These</span> <span m="1954480">things</span> <span m="1954700">are</span> <span m="1954820">not</span> <span m="1955350">immediately</span> <span m="1955870">obvious,</span> <span m="1956280">but</span> <span m="1956400">become</span> <span m="1956800">very</span> <span m="1957060">clear</span> <span m="1957400">once</span> <span m="1957640">you</span> <span m="1959050">know</span> <span m="1959150">the</span> <span m="1959220">mathematics</span> <span m="1959850">behind</span> <span m="1960170">it.</span> <span m="1962240">Any</span> <span m="1962420">questions?</span> <span m="1963860">Yeah.</span> </p>
<p><span m="1964645">AUDIENCE: How would</span> <span m="1965110">a person</span> <span m="1965575">hit a</span> <span m="1966040">random</span> <span m="1966505">number?</span> </p>
<p><span m="1967900">PROFESSOR: Oh,</span> <span m="1968390">you</span> <span m="1968610">go</span> <span m="1968790">to your</span> <span m="1968990">favorite</span> <span m="1969310">computer</span> <span m="1969870">and</span> <span m="1970460">do</span> <span m="1970620">a</span> <span m="1970710">random</span> <span m="1970980">number</span> <span m="1971210">generator.</span> <span m="1972290">Now,</span> <span m="1972530">that's</span> <span m="1972710">not</span> <span m="1972910">perfectly</span> <span m="1973370">random.</span> <span m="1974490">People</span> <span m="1974760">do</span> <span m="1974940">get</span> <span m="1975080">into</span> <span m="1975240">a</span> <span m="1975280">science</span> <span m="1975760">of</span> <span m="1975840">this</span> <span m="1976050">where</span> <span m="1976150">they're</span> <span m="1976310">certain</span> <span m="1977860">cosmic</span> <span m="1978380">rays</span> <span m="1978750">or</span> <span m="1978800">whatever</span> <span m="1979100">hitting</span> <span m="1979360">the</span> <span m="1979520">earth</span> <span m="1979790">and</span> <span m="1979880">the</span> <span m="1979940">frequency</span> <span m="1980470">they</span> <span m="1980600">try</span> <span m="1980770">to</span> <span m="1980850">get</span> <span m="1981020">random</span> <span m="1981290">numbers</span> <span m="1981610">out</span> <span m="1981780">of</span> <span m="1981870">it</span> <span m="1982010">or</span> <span m="1982550">certain</span> <span m="1982970">kinds</span> <span m="1983190">of</span> <span m="1983270">clocks</span> <span m="1983890">and</span> <span m="1984000">stuff</span> <span m="1984220">like</span> <span m="1984410">that,</span> <span m="1984760">the</span> <span m="1985250">tiny</span> <span m="1985650">low</span> <span m="1985830">order</span> <span m="1986020">bits.</span> </p>
<p><span m="1987090">Actually</span> <span m="1987530">getting</span> <span m="1988040">really</span> <span m="1988380">independent</span> <span m="1989530">random</span> <span m="1989840">numbers</span> <span m="1990080">can</span> <span m="1990340">be</span> <span m="1990430">challenging.</span> <span m="1990860">A</span> <span m="1990900">lot</span> <span m="1991030">of the</span> <span m="1991110">things</span> <span m="1991290">you</span> <span m="1991360">do</span> <span m="1991470">with</span> <span m="1991590">a</span> <span m="1992130">computer</span> <span m="1992710">generating</span> <span m="1993140">random</span> <span m="1993380">numbers,</span> <span m="1995060">they're</span> <span m="1995180">distributed</span> <span m="1995750">nicely,</span> <span m="1996140">but</span> <span m="1996250">they're</span> <span m="1996340">not</span> <span m="1996630">mutually</span> <span m="1997020">independent.</span> <span m="1997850">And</span> <span m="1997970">there's</span> <span m="1998190">whole</span> <span m="1998810">texts</span> <span m="1999250">that</span> <span m="1999360">go</span> <span m="1999550">into</span> <span m="1999730">how</span> <span m="1999880">to</span> <span m="1999970">do</span> <span m="2000250">that</span> <span m="2000670">for</span> <span m="2000800">mutual</span> <span m="2001240">independence.</span> <span m="2002810">Getting</span> <span m="2003050">something</span> <span m="2003230">that's</span> <span m="2003380">fair</span> <span m="2003750">for</span> <span m="2003860">one</span> <span m="2004060">of</span> <span m="2004120">them</span> <span m="2004260">is</span> <span m="2004350">not</span> <span m="2004820">too</span> <span m="2004990">hard.</span> <span m="2007100">Any</span> <span m="2007290">questions</span> <span m="2009110">about</span> <span m="2009400">that?</span> </p>
<p><span m="2010920">There's</span> <span m="2011090">another</span> <span m="2011320">example.</span> <span m="2012970">How</span> <span m="2013190">many</span> <span m="2013350">people</span> <span m="2013660">ever</span> <span m="2013910">participated</span> <span m="2014920">in</span> <span m="2015090">a</span> <span m="2015140">Super</span> <span m="2015510">Bowl</span> <span m="2016370">bet?</span> <span m="2018090">OK,</span> <span m="2018570">like</span> <span m="2018750">maybe</span> <span m="2018830">you're</span> <span m="2018920">trying</span> <span m="2019240">to</span> <span m="2019310">guess</span> <span m="2019630">the</span> <span m="2019760">over</span> <span m="2020060">under</span> <span m="2020410">on</span> <span m="2020530">point</span> <span m="2020810">scored.</span> <span m="2022930">And</span> <span m="2023420">the</span> <span m="2023790">person</span> <span m="2024160">who</span> <span m="2024300">guesses</span> <span m="2024810">close</span> <span m="2025340">to</span> <span m="2025430">the</span> <span m="2025510">total</span> <span m="2025780">number</span> <span m="2025950">of</span> <span m="2026030">points</span> <span m="2027050">wins</span> <span m="2027410">the</span> <span m="2027500">pot.</span> <span m="2028370">And</span> <span m="2028540">if</span> <span m="2028600">there's</span> <span m="2028740">a</span> <span m="2028790">tie,</span> <span m="2029190">it's</span> <span m="2029300">shared.</span> </p>
<p><span m="2030760">So</span> <span m="2030980">in</span> <span m="2031050">that</span> <span m="2031200">kind</span> <span m="2031350">of</span> <span m="2031420">situation,</span> <span m="2032520">some</span> <span m="2033180">people</span> <span m="2033490">figured</span> <span m="2033840">out</span> <span m="2034150">that</span> <span m="2034260">the</span> <span m="2034400">average</span> <span m="2034780">number</span> <span m="2035000">of</span> <span m="2035090">points</span> <span m="2035410">scored</span> <span m="2036570">in</span> <span m="2036710">a</span> <span m="2036750">Super</span> <span m="2037050">Bowl</span> <span m="2038520">is,</span> <span m="2038810">say,</span> <span m="2039040">30.</span> <span m="2040860">And</span> <span m="2042270">a</span> <span m="2042380">lot</span> <span m="2042590">of</span> <span m="2042660">the</span> <span m="2042740">guesses</span> <span m="2043230">then</span> <span m="2043570">will</span> <span m="2043690">cluster</span> <span m="2044240">around</span> <span m="2044720">30</span> <span m="2045010">points.</span> <span m="2047730">Now,</span> <span m="2048290">if</span> <span m="2048469">you</span> <span m="2048610">knew</span> <span m="2048960">this</span> <span m="2049610">and</span> <span m="2049780">you</span> <span m="2049860">knew</span> <span m="2050020">a</span> <span m="2050090">lot</span> <span m="2050290">of</span> <span m="2050350">the</span> <span m="2050429">guesses,</span> <span m="2050840">because</span> <span m="2051070">there</span> <span m="2051159">may</span> <span m="2051300">be</span> <span m="2051420">people</span> <span m="2051679">who</span> <span m="2051810">cared</span> <span m="2052370">betting,</span> <span m="2053010">where</span> <span m="2053409">would</span> <span m="2053550">you</span> <span m="2053639">make,</span> <span m="2054270">say,</span> <span m="2054510">this</span> <span m="2054730">is,</span> <span m="2054940">I</span> <span m="2055020">don't</span> <span m="2055120">know,</span> <span m="2055199">40</span> <span m="2055679">and</span> <span m="2055780">this</span> <span m="2055940">is</span> <span m="2056050">20,</span> <span m="2056560">where</span> <span m="2056929">would</span> <span m="2057080">you</span> <span m="2057219">guess?</span> </p>
<p><span m="2059440">You</span> <span m="2059580">could</span> <span m="2059730">guess</span> <span m="2060040">here.</span> <span m="2060480">That's</span> <span m="2060679">a</span> <span m="2060730">good</span> <span m="2060900">one.</span> <span m="2062120">Or</span> <span m="2062290">you</span> <span m="2062389">could</span> <span m="2062550">guess</span> <span m="2062850">here.</span> <span m="2064620">Not</span> <span m="2064870">only</span> <span m="2065070">are</span> <span m="2065150">you</span> <span m="2065260">not</span> <span m="2065469">going</span> <span m="2065590">to</span> <span m="2065670">share</span> <span m="2065920">the</span> <span m="2066010">pot,</span> <span m="2066460">which</span> <span m="2066670">helps</span> <span m="2066940">you,</span> <span m="2067699">but</span> <span m="2067750">in</span> <span m="2067889">this</span> <span m="2068100">case</span> <span m="2068440">you'd</span> <span m="2068570">actually,</span> <span m="2068920">because</span> <span m="2069150">being</span> <span m="2069360">closest,</span> <span m="2069739">you'd</span> <span m="2069880">capture</span> <span m="2070409">all</span> <span m="2070600">the</span> <span m="2070690">scores</span> <span m="2071110">out</span> <span m="2071270">here.</span> <span m="2072630">So</span> <span m="2072770">that's</span> <span m="2072989">really</span> <span m="2073260">good.</span> <span m="2074870">So</span> <span m="2074960">these</span> <span m="2075290">are</span> <span m="2075330">the</span> <span m="2075409">best</span> <span m="2075710">guesses</span> <span m="2076070">to</span> <span m="2076179">make</span> <span m="2076710">for</span> <span m="2076810">your</span> <span m="2076920">expected</span> <span m="2077320">return,</span> <span m="2078280">assuming</span> <span m="2078770">everybody</span> <span m="2079090">is</span> <span m="2079210">guessing</span> <span m="2079550">around</span> <span m="2079850">here</span> <span m="2080060">because</span> <span m="2080360">they</span> <span m="2080489">know</span> <span m="2080730">that's</span> <span m="2080989">the</span> <span m="2081510">median.</span> <span m="2081880">Yeah?</span> </p>
<p><span m="2082846">AUDIENCE: But those scores</span> <span m="2083329">aren't</span> <span m="2083812">likely to happen.</span> </p>
<p><span m="2085750">PROFESSOR: They're</span> <span m="2085909">not</span> <span m="2086290">likely</span> <span m="2086420">to</span> <span m="2086520">happen,</span> <span m="2087050">but it</span> <span m="2087199">can</span> <span m="2087449">outweigh</span> <span m="2089190">splitting</span> <span m="2089639">the</span> <span m="2089719">pot</span> <span m="2090199">with</span> <span m="2090360">all</span> <span m="2090739">the</span> <span m="2090820">people</span> <span m="2091159">that</span> <span m="2091250">guessed</span> <span m="2091659">here.</span> <span m="2093870">And</span> <span m="2094230">you</span> <span m="2094300">get</span> <span m="2094429">to</span> <span m="2094489">cover</span> <span m="2095360">more</span> <span m="2095570">bases.</span> <span m="2096949">So</span> <span m="2097150">you're</span> <span m="2097300">right,</span> <span m="2097560">they're</span> <span m="2097650">less</span> <span m="2097930">likely</span> <span m="2098180">to</span> <span m="2098280">happen</span> <span m="2098660">because</span> <span m="2099100">so</span> <span m="2099530">many</span> <span m="2099760">people</span> <span m="2100110">guessed</span> <span m="2100470">in</span> <span m="2100590">here</span> <span m="2100850">where</span> <span m="2101000">it's</span> <span m="2101140">likely.</span> <span m="2102170">Your</span> <span m="2102380">expected</span> <span m="2103010">return</span> <span m="2104270">is</span> <span m="2104440">better</span> <span m="2104750">out</span> <span m="2104950">here.</span> <span m="2105790">You</span> <span m="2105930">may</span> <span m="2106050">not</span> <span m="2106200">be</span> <span m="2106300">likely</span> <span m="2106570">to</span> <span m="2106650">win,</span> <span m="2108060">but</span> <span m="2108230">your</span> <span m="2108340">payoff</span> <span m="2108750">will</span> <span m="2108840">be</span> <span m="2108930">very</span> <span m="2109170">large</span> <span m="2109430">when</span> <span m="2109550">you</span> <span m="2109620">do.</span> </p>
<p><span m="2111790">Now,</span> <span m="2112250">if</span> <span m="2112490">you're</span> <span m="2112640">in</span> <span m="2112730">a</span> <span m="2113390">bet</span> <span m="2113980">or</span> <span m="2114140">a</span> <span m="2114200">pool</span> <span m="2114590">with</span> <span m="2114730">a</span> <span m="2114780">bunch</span> <span m="2115030">of</span> <span m="2115110">6042</span> <span m="2115860">students</span> <span m="2116880">and</span> <span m="2117060">they're</span> <span m="2117230">all</span> <span m="2117400">guessing</span> <span m="2117810">out</span> <span m="2117980">here,</span> <span m="2119210">well</span> <span m="2119610">then</span> <span m="2119780">you</span> <span m="2119850">want</span> <span m="2120010">to</span> <span m="2120070">go</span> <span m="2120190">back</span> <span m="2120430">home</span> <span m="2121150">here.</span> <span m="2122730">Then</span> <span m="2122880">it's</span> <span m="2123030">better.</span> <span m="2123250">So</span> <span m="2123340">it</span> <span m="2123410">all</span> <span m="2123590">depends</span> <span m="2123980">on</span> <span m="2124060">what</span> <span m="2124200">that</span> <span m="2124350">distribution</span> <span m="2125620">looks</span> <span m="2125910">like.</span> <span m="2128800">All</span> <span m="2128970">right,</span> <span m="2129110">any</span> <span m="2129680">more</span> <span m="2129910">questions</span> <span m="2130310">about</span> <span m="2130540">this?</span> <span m="2131660">Yeah.</span> </p>
<p><span m="2132980">AUDIENCE: [INAUDIBLE]</span> <span m="2133760">if you're</span> <span m="2134100">going</span> <span m="2134213">to</span> <span m="2134326">be</span> <span m="2134440">playing</span> <span m="2134680">a whole</span> <span m="2135156">bunch of</span> <span m="2135632">times,</span> <span m="2136108">and if you're</span> <span m="2136584">only</span> <span m="2137060">playing once,</span> <span m="2137540">wouldn't you look at</span> <span m="2137976">the most likely</span> <span m="2138412">result?</span> </p>
<p><span m="2140160">PROFESSOR: Say</span> <span m="2140500">that</span> <span m="2140630">again</span> <span m="2140950">now?</span> </p>
<p><span m="2141408">AUDIENCE: If you're</span> <span m="2141866">going to be</span> <span m="2142324">only</span> <span m="2142782">playing once</span> <span m="2143240">wouldn't</span> <span m="2143698">you</span> <span m="2143929">only be</span> <span m="2144160">concerned with</span> <span m="2144657">the result</span> <span m="2145154">that's</span> <span m="2145402">most</span> <span m="2145651">likely to</span> <span m="2146148">show up?</span> </p>
<p><span m="2147640">PROFESSOR: Well,</span> <span m="2148330">it</span> <span m="2148400">depends</span> <span m="2148730">on</span> <span m="2148800">your</span> <span m="2148920">strategy.</span> <span m="2149480">If</span> <span m="2149580">you</span> <span m="2149690">want</span> <span m="2149940">the</span> <span m="2150080">expected</span> <span m="2150770">gain</span> <span m="2151170">to</span> <span m="2151240">be</span> <span m="2151330">maximized,</span> <span m="2152580">it</span> <span m="2152650">doesn't</span> <span m="2152860">matter</span> <span m="2153080">how</span> <span m="2153220">many</span> <span m="2153380">times</span> <span m="2153630">you're</span> <span m="2153720">playing.</span> <span m="2154760">And</span> <span m="2154920">that</span> <span m="2155090">can</span> <span m="2155210">be</span> <span m="2155310">very</span> <span m="2155590">different</span> <span m="2156550">than</span> <span m="2156760">maximizing</span> <span m="2157400">your</span> <span m="2157500">probability</span> <span m="2158050">of</span> <span m="2158150">winning.</span> <span m="2159480">If</span> <span m="2159640">you</span> <span m="2159710">want</span> <span m="2159860">to</span> <span m="2159920">maximize</span> <span m="2160420">your</span> <span m="2160510">probability</span> <span m="2160990">of</span> <span m="2161080">winning,</span> <span m="2162210">you're</span> <span m="2162370">going</span> <span m="2162490">to</span> <span m="2162560">go</span> <span m="2162870">right</span> <span m="2163240">at</span> <span m="2164270">the</span> <span m="2164420">center</span> <span m="2164700">point</span> <span m="2165010">here.</span> </p>
<p><span m="2166320">Because</span> <span m="2166500">if</span> <span m="2166620">you</span> <span m="2166730">look</span> <span m="2166890">at</span> <span m="2166950">the</span> <span m="2167020">probability</span> <span m="2167550">distribution</span> <span m="2168290">and</span> <span m="2168420">say</span> <span m="2168570">the</span> <span m="2168650">probability</span> <span m="2169070">distribution</span> <span m="2169650">looks</span> <span m="2169880">like</span> <span m="2170800">that,</span> <span m="2172670">then</span> <span m="2173000">you</span> <span m="2173100">want</span> <span m="2173270">to</span> <span m="2173330">bet</span> <span m="2173720">here,</span> <span m="2173980">because</span> <span m="2174130">that's</span> <span m="2174280">the</span> <span m="2174350">maximum</span> <span m="2174790">chance</span> <span m="2175120">of</span> <span m="2175210">winning.</span> <span m="2177390">But</span> <span m="2177590">say</span> <span m="2177790">so</span> <span m="2178160">many</span> <span m="2178370">people</span> <span m="2178660">bet</span> <span m="2178910">there,</span> <span m="2179040">you'd</span> <span m="2179210">split</span> <span m="2180250">30</span> <span m="2180720">ways.</span> <span m="2182250">Well,</span> <span m="2182500">this</span> <span m="2182820">divided</span> <span m="2183180">by</span> <span m="2183310">30</span> <span m="2184270">is</span> <span m="2184420">smaller</span> <span m="2184910">than</span> <span m="2185090">this</span> <span m="2185500">divided</span> <span m="2185860">by</span> <span m="2185990">1.</span> <span m="2187260">And</span> <span m="2187420">so</span> <span m="2187510">your</span> <span m="2187610">expected</span> <span m="2188060">return</span> <span m="2188370">will</span> <span m="2188460">be</span> <span m="2188520">bigger</span> <span m="2188780">out</span> <span m="2188930">here.</span> <span m="2191280">So</span> <span m="2191480">two</span> <span m="2191790">different</span> <span m="2192130">things.</span> <span m="2192500">Maximizing</span> <span m="2193060">the</span> <span m="2193170">probability</span> <span m="2193550">of</span> <span m="2193620">winning</span> <span m="2194190">and</span> <span m="2194400">maximizing</span> <span m="2195160">your</span> <span m="2195270">expected</span> <span m="2195690">return.</span> <span m="2196830">Yeah?</span> </p>
<p><span m="2197820">AUDIENCE: When  you say</span> <span m="2198315">maximizing,</span> <span m="2198810">do</span> <span m="2199305">you just</span> <span m="2199800">do</span> <span m="2200295">derivatives?</span> </p>
<p><span m="2201780">PROFESSOR: Yeah,</span> <span m="2201990">then</span> <span m="2202240">you</span> <span m="2202340">could</span> <span m="2202450">do</span> <span m="2202610">derivatives</span> <span m="2203060">and</span> <span m="2203150">all</span> <span m="2203270">that</span> <span m="2203430">kind</span> <span m="2203550">of</span> <span m="2203630">stuff.</span> <span m="2203840">That's</span> <span m="2204070">right.</span> <span m="2204350">Once</span> <span m="2204560">you</span> <span m="2204650">have</span> <span m="2204800">the</span> <span m="2205220">curve,</span> <span m="2205400">you figure</span> <span m="2205480">it out.</span> <span m="2205948">Yeah.</span> <span m="2207820">Yeah.</span> </p>
<p><span m="2208551">AUDIENCE: Wouldn't it</span> <span m="2209032">be better</span> <span m="2209513">to maximize</span> <span m="2209994">your</span> <span m="2212880">expected</span> <span m="2213361">value</span> <span m="2213842">versus</span> <span m="2214323">the probability</span> <span m="2214804">that you</span> <span m="2215285">win?</span> </p>
<p><span m="2216260">PROFESSOR: Yes.</span> <span m="2216740">In</span> <span m="2216860">general,</span> <span m="2217230">you</span> <span m="2217370">want</span> <span m="2217720">to</span> <span m="2217780">maximize</span> <span m="2219070">the</span> <span m="2219200">expected</span> <span m="2219720">return</span> <span m="2221160">on</span> <span m="2221380">the</span> <span m="2221450">basis</span> <span m="2221950">that</span> <span m="2222100">probably</span> <span m="2222530">over</span> <span m="2222730">life</span> <span m="2223020">you're</span> <span m="2223170">doing</span> <span m="2223440">lots</span> <span m="2223720">of</span> <span m="2223800">things.</span> <span m="2224950">And</span> <span m="2225030">that</span> <span m="2225270">overall</span> <span m="2225870">puts</span> <span m="2226070">you</span> <span m="2226170">in</span> <span m="2226240">a</span> <span m="2226320">better</span> <span m="2226590">state.</span> <span m="2226990">Now,</span> <span m="2227930">we're</span> <span m="2227990">going</span> <span m="2228110">to</span> <span m="2228170">talk</span> <span m="2228380">about</span> <span m="2228620">this</span> <span m="2229110">some</span> <span m="2229390">next</span> <span m="2229780">time</span> <span m="2232431">in</span> <span m="2232860">terms</span> <span m="2233270">of</span> <span m="2233670">taking</span> <span m="2233980">high</span> <span m="2234330">risk</span> <span m="2234750">bets</span> <span m="2235130">with</span> <span m="2235250">high,</span> <span m="2235636">high</span> <span m="2236410">payoffs.</span> <span m="2237860">That</span> <span m="2238080">can</span> <span m="2238230">maximize</span> <span m="2238820">your</span> <span m="2238930">expected</span> <span m="2239360">return,</span> <span m="2240370">but</span> <span m="2240590">you</span> <span m="2240650">have</span> <span m="2240750">a</span> <span m="2240790">very</span> <span m="2241060">decent</span> <span m="2241390">chance</span> <span m="2241700">of</span> <span m="2241800">losing</span> <span m="2242080">a</span> <span m="2242140">lot.</span> </p>
<p><span m="2243230">And</span> <span m="2243430">so</span> <span m="2243530">you</span> <span m="2243610">might</span> <span m="2243870">not</span> <span m="2244070">want</span> <span m="2244210">to</span> <span m="2244270">go</span> <span m="2244470">there.</span> <span m="2244840">And</span> <span m="2244970">we'll</span> <span m="2245070">talk</span> <span m="2245280">about</span> <span m="2245490">that</span> <span m="2245650">next</span> <span m="2245850">time</span> <span m="2246020">we</span> <span m="2246120">talk</span> <span m="2246320">about</span> <span m="2246510">variance</span> <span m="2247060">and</span> <span m="2247570">actually</span> <span m="2248320">what</span> <span m="2248640">your</span> <span m="2248780">choice</span> <span m="2249110">is.</span> <span m="2249790">Because</span> <span m="2250020">that</span> <span m="2250170">is</span> <span m="2250330">a</span> <span m="2250380">fundamental</span> <span m="2250820">choice</span> <span m="2251110">you</span> <span m="2251460">face.</span> <span m="2252110">Maximizing</span> <span m="2252810">chance</span> <span m="2253100">of</span> <span m="2253180">winning,</span> <span m="2253840">maximizing</span> <span m="2254460">expected</span> <span m="2254870">return.</span> <span m="2255760">And</span> <span m="2255850">of</span> <span m="2255910">course,</span> <span m="2256100">tied</span> <span m="2256380">into</span> <span m="2256560">that</span> <span m="2256740">is</span> <span m="2256840">the</span> <span m="2256950">risk</span> <span m="2257440">of</span> <span m="2257580">losing</span> <span m="2257970">and</span> <span m="2258110">what</span> <span m="2258250">kind</span> <span m="2258380">of</span> <span m="2258460">risk</span> <span m="2258730">you're</span> <span m="2258820">willing</span> <span m="2259020">to</span> <span m="2259080">tolerate.</span> </p>
<p><span m="2263530">We're</span> <span m="2263740">going</span> <span m="2263870">to</span> <span m="2263930">do</span> <span m="2264040">a</span> <span m="2264100">bunch</span> <span m="2264320">more</span> <span m="2264460">examples,</span> <span m="2265130">but</span> <span m="2265150">before</span> <span m="2265470">I</span> <span m="2265550">do,</span> <span m="2267050">I</span> <span m="2267150">want</span> <span m="2267170">to</span> <span m="2267250">show</span> <span m="2267440">you</span> <span m="2267540">some</span> <span m="2267760">other</span> <span m="2268160">equivalent</span> <span m="2268810">definitions</span> <span m="2269470">of</span> <span m="2269570">expectation.</span> <span m="2283650">So</span> <span m="2283720">the</span> <span m="2283830">expected</span> <span m="2284300">value</span> <span m="2284560">of</span> <span m="2284620">a</span> <span m="2284700">random</span> <span m="2285030">variable,</span> <span m="2286150">you</span> <span m="2286170">can</span> <span m="2286300">also</span> <span m="2287110">compute</span> <span m="2287530">it by</span> <span m="2288270">summing</span> <span m="2288900">over</span> <span m="2289290">all</span> <span m="2289670">possible</span> <span m="2290150">values</span> <span m="2290600">of</span> <span m="2290660">the</span> <span m="2290750">random</span> <span m="2291070">variable.</span> <span m="2292400">x</span> <span m="2292730">being</span> <span m="2292900">in</span> <span m="2292970">the</span> <span m="2293070">range</span> <span m="2293390">of</span> <span m="2293510">R.</span> <span m="2296010">x</span> <span m="2296490">times</span> <span m="2296910">the</span> <span m="2296980">probability</span> <span m="2297600">that</span> <span m="2297720">R</span> <span m="2297980">equals</span> <span m="2298390">x.</span> </p>
<p><span m="2300330">And</span> <span m="2300530">let's</span> <span m="2300700">see</span> <span m="2300850">why</span> <span m="2301070">this</span> <span m="2301270">is</span> <span m="2301360">true.</span> <span m="2302200">It</span> <span m="2302420">follows</span> <span m="2303300">from</span> <span m="2303550">the</span> <span m="2303620">definition.</span> <span m="2307600">From</span> <span m="2307810">the</span> <span m="2307890">definition,</span> <span m="2308520">we</span> <span m="2308630">know</span> <span m="2308850">the</span> <span m="2308930">expected</span> <span m="2309370">value</span> <span m="2310270">is</span> <span m="2310440">the</span> <span m="2310530">sum</span> <span m="2310910">over</span> <span m="2311150">all</span> <span m="2311330">sample</span> <span m="2311760">points</span> <span m="2313600">of</span> <span m="2313910">the</span> <span m="2313990">value</span> <span m="2314460">of</span> <span m="2314520">the</span> <span m="2314620">random</span> <span m="2314890">variable</span> <span m="2315320">on</span> <span m="2315530">the</span> <span m="2315600">sample</span> <span m="2315970">point</span> <span m="2316580">times</span> <span m="2316950">its</span> <span m="2317040">probability.</span> <span m="2319970">And</span> <span m="2320120">now</span> <span m="2320300">we</span> <span m="2320430">can</span> <span m="2321000">organize</span> <span m="2321700">this</span> <span m="2321900">sum</span> <span m="2323020">by</span> <span m="2323250">the</span> <span m="2323350">value</span> <span m="2324880">that</span> <span m="2325030">R</span> <span m="2325270">takes.</span> </p>
<p><span m="2326880">So</span> <span m="2327090">we're</span> <span m="2327160">going</span> <span m="2327280">to</span> <span m="2327350">split</span> <span m="2327630">this</span> <span m="2327800">into</span> <span m="2327980">a</span> <span m="2328040">double</span> <span m="2328370">sum</span> <span m="2330480">where</span> <span m="2330770">here</span> <span m="2331090">we're</span> <span m="2331220">looking</span> <span m="2331530">first</span> <span m="2332670">at</span> <span m="2332890">x</span> <span m="2333160">in</span> <span m="2333230">the</span> <span m="2333340">range</span> <span m="2333610">of</span> <span m="2333740">R.</span> <span m="2337330">And</span> <span m="2337510">then</span> <span m="2337640">here</span> <span m="2338000">we</span> <span m="2338130">look</span> <span m="2338360">at</span> <span m="2338800">sample</span> <span m="2339290">points</span> <span m="2339700">for</span> <span m="2339820">which</span> <span m="2341690">R</span> <span m="2342030">on</span> <span m="2342140">that</span> <span m="2342320">point</span> <span m="2342770">is</span> <span m="2343000">x.</span> <span m="2344380">So</span> <span m="2344610">this</span> <span m="2344850">sum</span> <span m="2345120">is</span> <span m="2345230">equivalent</span> <span m="2345870">to</span> <span m="2345960">this</span> <span m="2346210">one.</span> <span m="2347220">Here</span> <span m="2347410">we're</span> <span m="2347500">just</span> <span m="2347730">organizing</span> <span m="2348370">all</span> <span m="2348510">the</span> <span m="2348590">sample</span> <span m="2348970">points</span> <span m="2349260">for</span> <span m="2349330">the</span> <span m="2349400">same</span> <span m="2349680">value</span> <span m="2349990">of</span> <span m="2350090">x.</span> </p>
<p><span m="2368990">All</span> <span m="2369140">right,</span> <span m="2369300">now</span> <span m="2370680">in</span> <span m="2370790">the</span> <span m="2370940">inner</span> <span m="2371200">sum,</span> <span m="2372220">I've</span> <span m="2372430">only</span> <span m="2372680">got</span> <span m="2373650">values</span> <span m="2374110">for</span> <span m="2374210">which</span> <span m="2374430">R</span> <span m="2374660">w</span> <span m="2374950">equals</span> <span m="2375260">x,</span> <span m="2375580">so</span> <span m="2375680">I</span> <span m="2375740">can</span> <span m="2375900">just</span> <span m="2376140">replace</span> <span m="2376570">this</span> <span m="2376790">with</span> <span m="2376950">x.</span> <span m="2390710">The</span> <span m="2390770">same</span> <span m="2390880">as</span> <span m="2391010">before.</span> <span m="2393860">Only</span> <span m="2394050">now</span> <span m="2394230">I've</span> <span m="2394660">just</span> <span m="2394880">put</span> <span m="2395390">x</span> <span m="2395680">instead</span> <span m="2396050">of</span> <span m="2396180">Rw.</span> <span m="2401750">Now</span> <span m="2401970">I</span> <span m="2402050">can</span> <span m="2402230">pull</span> <span m="2402620">the</span> <span m="2402770">x</span> <span m="2403190">out</span> <span m="2403770">since</span> <span m="2404060">it's</span> <span m="2404210">a</span> <span m="2404530">constant</span> <span m="2405540">independent</span> <span m="2405805">of</span> <span m="2406070">that</span> <span m="2406240">sum.</span> </p>
<p><span m="2423070">Now</span> <span m="2423310">here</span> <span m="2423670">I'm</span> <span m="2423790">summing</span> <span m="2424260">up</span> <span m="2425400">the</span> <span m="2425540">probability</span> <span m="2426200">of</span> <span m="2426290">all</span> <span m="2426700">the</span> <span m="2426800">sample</span> <span m="2427280">points</span> <span m="2427700">for</span> <span m="2427810">which</span> <span m="2428180">R</span> <span m="2429630">of</span> <span m="2429760">the</span> <span m="2429840">sample</span> <span m="2430160">point</span> <span m="2430440">is</span> <span m="2430550">x.</span> <span m="2432100">And</span> <span m="2432270">that</span> <span m="2433440">is</span> <span m="2433690">just</span> <span m="2435050">the</span> <span m="2435160">probability</span> <span m="2435890">that</span> <span m="2436030">R</span> <span m="2436230">equals</span> <span m="2436520">x.</span> <span m="2437890">That's</span> <span m="2438130">the</span> <span m="2438200">definition</span> <span m="2438730">of</span> <span m="2438790">the</span> <span m="2438850">probability</span> <span m="2439260">of that</span> <span m="2439430">event.</span> <span m="2440540">And</span> <span m="2440730">so</span> <span m="2440950">now</span> <span m="2441160">my</span> <span m="2441380">answer</span> <span m="2441750">is</span> <span m="2442550">some</span> <span m="2442820">overall</span> <span m="2443720">values</span> <span m="2444310">x</span> <span m="2444520">in</span> <span m="2444590">the</span> <span m="2444680">range</span> <span m="2446000">of</span> <span m="2446170">x</span> <span m="2446430">times</span> <span m="2446810">of</span> <span m="2446880">probability</span> <span m="2447330">R</span> <span m="2447540">equals</span> <span m="2447880">x.</span> <span m="2448350">And</span> <span m="2448510">that's</span> <span m="2448760">what</span> <span m="2448950">I</span> <span m="2448980">was</span> <span m="2449120">trying</span> <span m="2449320">to</span> <span m="2449390">prove.</span> </p>
<p><span m="2455870">OK,</span> <span m="2456050">any</span> <span m="2456180">questions</span> <span m="2458558">about</span> <span m="2458980">that?</span> <span m="2460990">Now</span> <span m="2461100">there</span> <span m="2461200">are</span> <span m="2461230">some</span> <span m="2461360">special</span> <span m="2461780">cases</span> <span m="2462270">of</span> <span m="2462370">this</span> <span m="2463230">when</span> <span m="2464050">the</span> <span m="2464200">random</span> <span m="2464560">variable</span> <span m="2465840">is</span> <span m="2466190">on</span> <span m="2466400">the</span> <span m="2466730">natural</span> <span m="2467180">numbers.</span> <span m="2468000">So</span> <span m="2468070">the</span> <span m="2468160">range</span> <span m="2468550">of</span> <span m="2468690">R is</span> <span m="2468970">the</span> <span m="2469030">natural</span> <span m="2469370">numbers.</span> </p>
<p><span m="2481960">So</span> <span m="2482130">a</span> <span m="2482170">corollary.</span> <span m="2485400">If</span> <span m="2485690">the</span> <span m="2485820">random</span> <span m="2486200">variable</span> <span m="2488640">has</span> <span m="2488820">a</span> <span m="2488910">range</span> <span m="2489260">on</span> <span m="2489330">the</span> <span m="2489400">natural</span> <span m="2489780">numbers,</span> <span m="2493400">then</span> <span m="2493640">another</span> <span m="2494000">way</span> <span m="2494150">to</span> <span m="2494220">compute</span> <span m="2494610">the</span> <span m="2494700">expected</span> <span m="2495150">value</span> <span m="2495510">of</span> <span m="2495630">r</span> <span m="2497560">is</span> <span m="2497780">to</span> <span m="2497870">simply</span> <span m="2498330">sum</span> <span m="2499730">i</span> <span m="2500000">equals</span> <span m="2500360">1</span> <span m="2500690">to</span> <span m="2500860">infinity</span> <span m="2502420">i</span> <span m="2503290">times</span> <span m="2503640">the</span> <span m="2503710">probability</span> <span m="2504390">R</span> <span m="2504820">equals</span> <span m="2505290">i.</span> <span m="2508330">And</span> <span m="2508450">the</span> <span m="2508520">proof,</span> <span m="2509370">well,</span> <span m="2509610">it's</span> <span m="2509920">really</span> <span m="2510200">just</span> <span m="2510400">saying</span> <span m="2510670">the</span> <span m="2510730">same</span> <span m="2511040">thing</span> <span m="2511190">here.</span> <span m="2511890">I'm</span> <span m="2511960">just</span> <span m="2512180">summing</span> <span m="2512470">over</span> <span m="2512630">the</span> <span m="2512690">natural</span> <span m="2513050">numbers.</span> <span m="2514160">And</span> <span m="2514320">the</span> <span m="2514400">case</span> <span m="2514800">with</span> <span m="2514950">0</span> <span m="2515510">doesn't</span> <span m="2515830">matter</span> <span m="2516140">because</span> <span m="2516310">I</span> <span m="2516380">get</span> <span m="2516580">0</span> <span m="2517120">times</span> <span m="2518170">the</span> <span m="2518240">probability</span> <span m="2518610">of 0.</span> <span m="2518830">And</span> <span m="2518930">that</span> <span m="2519170">adds</span> <span m="2519470">nothing</span> <span m="2519790">to</span> <span m="2519860">the</span> <span m="2519950">sum.</span> <span m="2522390">So</span> <span m="2522580">I</span> <span m="2522780">just have to</span> <span m="2522990">sum</span> <span m="2523270">over</span> <span m="2523450">the</span> <span m="2523940">positive</span> <span m="2524940">integers</span> <span m="2526110">in that</span> <span m="2526514">case.</span> </p>
<p><span m="2529390">All</span> <span m="2529530">right,</span> <span m="2529660">there's</span> <span m="2529800">another</span> <span m="2530840">special</span> <span m="2531290">case</span> <span m="2531620">of</span> <span m="2531750">this,</span> <span m="2533630">which</span> <span m="2534060">makes</span> <span m="2534340">it</span> <span m="2534410">even</span> <span m="2534630">easier</span> <span m="2535060">sometimes</span> <span m="2535540">to</span> <span m="2535620">compute</span> <span m="2536980">the</span> <span m="2537090">expected</span> <span m="2537345">value.</span> <span m="2559910">If</span> <span m="2560260">R</span> <span m="2561460">is</span> <span m="2561580">a</span> <span m="2561660">random</span> <span m="2561940">variable</span> <span m="2562320">in</span> <span m="2562380">the</span> <span m="2562450">natural</span> <span m="2562840">numbers,</span> <span m="2566260">then</span> <span m="2567930">the</span> <span m="2568070">expected</span> <span m="2568580">value</span> <span m="2568920">of</span> <span m="2569040">R</span> <span m="2570110">is</span> <span m="2570250">simply</span> <span m="2570570">the</span> <span m="2570660">sum</span> <span m="2572170">i</span> <span m="2572590">equals</span> <span m="2572930">0</span> <span m="2573690">to</span> <span m="2573830">infinity</span> <span m="2575320">of</span> <span m="2575400">the</span> <span m="2575500">probability</span> <span m="2576240">the</span> <span m="2576350">random</span> <span m="2576680">variable</span> <span m="2577070">is</span> <span m="2577200">bigger</span> <span m="2577590">than</span> <span m="2577750">i.</span> <span m="2580400">Which</span> <span m="2580740">is</span> <span m="2581750">the</span> <span m="2581830">same</span> <span m="2582260">as</span> <span m="2582560">summing</span> <span m="2583650">i</span> <span m="2584120">equals</span> <span m="2584310">1</span> <span m="2584600">to</span> <span m="2584730">infinity,</span> <span m="2585755">the</span> <span m="2586140">probability</span> <span m="2586960">R</span> <span m="2587340">is</span> <span m="2587680">bigger</span> <span m="2587950">than</span> <span m="2588120">or</span> <span m="2588210">equal</span> <span m="2588470">to</span> <span m="2588580">i.</span> <span m="2589300">They're</span> <span m="2589530">the</span> <span m="2589600">same</span> <span m="2589990">thing.</span> </p>
<p><span m="2591390">For</span> <span m="2591510">example,</span> <span m="2591780">the</span> <span m="2591840">first</span> <span m="2592150">term</span> <span m="2592370">here,</span> <span m="2592580">your</span> <span m="2592690">probability</span> <span m="2593160">R</span> <span m="2593410">bigger</span> <span m="2593670">than</span> <span m="2593800">0.</span> <span m="2594750">That's</span> <span m="2594960">the</span> <span m="2595030">same</span> <span m="2595330">as</span> <span m="2595430">saying</span> <span m="2595950">R</span> <span m="2596230">is</span> <span m="2596640">bigger</span> <span m="2596910">than or</span> <span m="2596970">equal</span> <span m="2597080">to</span> <span m="2597180">1</span> <span m="2598510">and</span> <span m="2598680">so</span> <span m="2598860">forth.</span> <span m="2599220">So</span> <span m="2599230">these</span> <span m="2599430">are</span> <span m="2599490">clearly</span> <span m="2599810">the</span> <span m="2599900">same.</span> <span m="2602000">And</span> <span m="2603250">the</span> <span m="2603350">difference</span> <span m="2603790">here</span> <span m="2604070">is</span> <span m="2604190">we</span> <span m="2604300">have</span> <span m="2605065">i</span> <span m="2605380">times</span> <span m="2605760">probability</span> <span m="2606140">R</span> <span m="2606290">equals</span> <span m="2606775">i.</span> <span m="2608230">Here</span> <span m="2608400">we</span> <span m="2608500">have</span> <span m="2608670">probability</span> <span m="2609080">R</span> <span m="2609350">is</span> <span m="2609520">bigger</span> <span m="2609870">than</span> <span m="2609990">I,</span> <span m="2610990">with</span> <span m="2611130">no</span> <span m="2611370">i</span> <span m="2611570">out</span> <span m="2611710">in</span> <span m="2611770">front.</span> <span m="2612120">So</span> <span m="2612210">let's</span> <span m="2612360">see</span> <span m="2612480">why</span> <span m="2612640">that's</span> <span m="2612790">true.</span> </p>
<p><span m="2637200">Well,</span> <span m="2637390">we're</span> <span m="2637460">going</span> <span m="2637580">to</span> <span m="2637930">work</span> <span m="2638220">backwards</span> <span m="2638930">and</span> <span m="2639170">evaluate</span> <span m="2640460">that</span> <span m="2641060">sum.</span> <span m="2642830">The</span> <span m="2642920">sum</span> <span m="2644580">i</span> <span m="2644990">equal</span> <span m="2645410">0</span> <span m="2646140">to</span> <span m="2646270">infinity</span> <span m="2648820">probability</span> <span m="2649440">R</span> <span m="2650110">is</span> <span m="2650520">bigger</span> <span m="2650870">than</span> <span m="2651010">i.</span> <span m="2653350">Well,</span> <span m="2654390">that's</span> <span m="2654860">the</span> <span m="2654950">probability</span> <span m="2656500">R</span> <span m="2656890">is</span> <span m="2657000">bigger</span> <span m="2657250">than</span> <span m="2657410">0</span> <span m="2658810">plus</span> <span m="2659190">the</span> <span m="2659280">probability</span> <span m="2660740">R</span> <span m="2661050">is</span> <span m="2661230">bigger</span> <span m="2661450">than</span> <span m="2661610">1</span> <span m="2663060">plus</span> <span m="2663440">probability</span> <span m="2664830">R</span> <span m="2665180">is</span> <span m="2665260">bigger</span> <span m="2665490">than</span> <span m="2665640">2,</span> <span m="2667700">and</span> <span m="2667870">so</span> <span m="2668070">forth</span> <span m="2668440">out</span> <span m="2668610">to</span> <span m="2668710">infinity.</span> <span m="2672610">Adding</span> <span m="2672930">those</span> <span m="2673160">up.</span> </p>
<p><span m="2675340">And</span> <span m="2675520">now</span> <span m="2675750">I</span> <span m="2675820">can</span> <span m="2676000">write</span> <span m="2676240">this</span> <span m="2676450">one</span> <span m="2676600">out.</span> <span m="2676910">That's</span> <span m="2677170">the</span> <span m="2677250">probability</span> <span m="2679190">R</span> <span m="2679520">equals</span> <span m="2679850">1</span> <span m="2680980">plus</span> <span m="2681430">the</span> <span m="2681570">probability</span> <span m="2682850">R</span> <span m="2683250">equals</span> <span m="2683690">2</span> <span m="2684550">plus</span> <span m="2684890">the</span> <span m="2684980">probability</span> <span m="2686430">R</span> <span m="2686830">equals</span> <span m="2687300">3</span> <span m="2688960">and</span> <span m="2689080">so</span> <span m="2689290">forth.</span> <span m="2692450">Probability</span> <span m="2692930">are</span> <span m="2693220">bigger</span> <span m="2693560">than</span> <span m="2693740">1</span> <span m="2694110">equals,</span> <span m="2695100">well,</span> <span m="2695500">R</span> <span m="2695860">could</span> <span m="2696210">be</span> <span m="2696310">2,</span> <span m="2698110">R</span> <span m="2698410">could</span> <span m="2698480">be</span> <span m="2698610">3,</span> <span m="2701730">and</span> <span m="2701900">so</span> <span m="2702100">forth.</span> </p>
<p><span m="2703760">R</span> <span m="2704110">bigger</span> <span m="2704360">than</span> <span m="2704540">2.</span> <span m="2705460">Well,</span> <span m="2705760">R</span> <span m="2705960">could</span> <span m="2706090">be</span> <span m="2706180">3,</span> <span m="2709410">4,</span> <span m="2709920">and</span> <span m="2710040">so</span> <span m="2710250">forth.</span> <span m="2712010">And</span> <span m="2712150">so</span> <span m="2712250">now</span> <span m="2712440">if</span> <span m="2712530">I</span> <span m="2712640">add</span> <span m="2712930">all</span> <span m="2713330">these</span> <span m="2713640">up,</span> <span m="2718110">well,</span> <span m="2718340">I</span> <span m="2718390">get</span> <span m="2718560">1</span> <span m="2720120">times</span> <span m="2720430">probability</span> <span m="2721610">R</span> <span m="2721910">equals</span> <span m="2722270">1.</span> <span m="2723780">Two</span> <span m="2724100">of</span> <span m="2724170">these</span> <span m="2724410">guys.</span> <span m="2730810">Three</span> <span m="2731180">of</span> <span m="2731270">these</span> <span m="2731530">guys.</span> <span m="2732400">And</span> <span m="2732600">you</span> <span m="2732680">can</span> <span m="2732800">see</span> <span m="2733070">the</span> <span m="2733190">pattern</span> <span m="2733620">here.</span> <span m="2736970">Before</span> <span m="2737540">the</span> <span m="2737640">next</span> <span m="2738190">and</span> <span m="2738330">so</span> <span m="2738540">forth.</span> <span m="2739810">And</span> <span m="2739970">so</span> <span m="2740080">we've</span> <span m="2740230">shown</span> <span m="2740650">that</span> <span m="2741910">this</span> <span m="2742260">value</span> <span m="2743640">equals</span> <span m="2744080">that</span> <span m="2744360">value,</span> <span m="2744850">which</span> <span m="2745150">is</span> <span m="2745390">by</span> <span m="2745580">the</span> <span m="2745700">corollary</span> <span m="2747830">just</span> <span m="2748160">the</span> <span m="2748260">expected</span> <span m="2748710">value</span> <span m="2748970">of</span> <span m="2749090">R.</span> <span m="2754900">So</span> <span m="2755130">by the</span> <span m="2755210">corollary,</span> <span m="2755640">we</span> <span m="2755740">know</span> <span m="2755900">that</span> <span m="2756020">the</span> <span m="2756080">expected</span> <span m="2756500">value</span> <span m="2756710">of</span> <span m="2756810">R</span> <span m="2757010">equals</span> <span m="2758160">the</span> <span m="2758280">sum</span> <span m="2759200">up</span> <span m="2759400">there.</span> <span m="2760450">And</span> <span m="2760540">that's</span> <span m="2760730">the</span> <span m="2760800">proof</span> <span m="2761080">of</span> <span m="2761150">the</span> <span m="2761230">theorem.</span> </p>
<p><span m="2766900">Sometimes</span> <span m="2767530">it's</span> <span m="2767690">easier</span> <span m="2768550">to</span> <span m="2768700">use</span> <span m="2769060">that</span> <span m="2769450">top</span> <span m="2769730">expression</span> <span m="2770210">there.</span> <span m="2771280">And</span> <span m="2772310">as</span> <span m="2772450">a</span> <span m="2772520">good</span> <span m="2772730">example,</span> <span m="2773920">it</span> <span m="2773990">gives</span> <span m="2774150">a</span> <span m="2774230">really</span> <span m="2774540">easy</span> <span m="2774900">way</span> <span m="2775610">to</span> <span m="2775740">compute</span> <span m="2776140">the</span> <span m="2776200">mean</span> <span m="2776520">time</span> <span m="2776820">to</span> <span m="2776900">failure</span> <span m="2777730">in</span> <span m="2777860">a</span> <span m="2777900">system.</span> <span m="2779690">So</span> <span m="2779720">let's</span> <span m="2779910">do</span> <span m="2780090">that.</span> </p>
<p><span m="2790290">Suppose</span> <span m="2790720">you</span> <span m="2790820">have</span> <span m="2791000">a</span> <span m="2791040">system</span> <span m="2795540">and</span> <span m="2795740">it</span> <span m="2795870">fails</span> <span m="2798390">with</span> <span m="2798700">probability</span> <span m="2799320">p</span> <span m="2802020">at</span> <span m="2802210">each</span> <span m="2802410">step.</span> <span m="2805210">And</span> <span m="2805470">let's</span> <span m="2805650">assume</span> <span m="2806180">that</span> <span m="2806310">the</span> <span m="2806380">failures</span> <span m="2807160">are</span> <span m="2807350">mutually</span> <span m="2807900">independent.</span> <span m="2810500">So</span> <span m="2810780">if</span> <span m="2810870">the</span> <span m="2810960">system</span> <span m="2811340">has</span> <span m="2811430">been</span> <span m="2811600">going</span> <span m="2811870">for</span> <span m="2811990">t</span> <span m="2812260">steps,</span> <span m="2813340">it</span> <span m="2813520">still</span> <span m="2813850">will</span> <span m="2813970">fail</span> <span m="2814320">on</span> <span m="2814460">step</span> <span m="2814770">t</span> <span m="2814940">plus</span> <span m="2815260">1</span> <span m="2815490">with</span> <span m="2815600">probability</span> <span m="2816140">p,</span> <span m="2816700">no</span> <span m="2816800">matter</span> <span m="2817010">what's</span> <span m="2817320">happened</span> <span m="2817640">before.</span> </p>
<p><span m="2819380">And</span> <span m="2819430">the</span> <span m="2819490">question</span> <span m="2819920">is,</span> <span m="2820270">what's</span> <span m="2820470">the</span> <span m="2820590">expected</span> <span m="2821420">number</span> <span m="2821670">of</span> <span m="2821760">steps</span> <span m="2822950">before</span> <span m="2823290">the</span> <span m="2823380">system</span> <span m="2823730">fails?</span> <span m="2824330">How</span> <span m="2824530">long</span> <span m="2824740">is</span> <span m="2824850">it</span> <span m="2824910">going</span> <span m="2825030">to</span> <span m="2825100">live</span> <span m="2825630">before</span> <span m="2825970">you</span> <span m="2826050">get</span> <span m="2826350">a</span> <span m="2826440">crash?</span> <span m="2827680">And</span> <span m="2827860">we're</span> <span m="2827930">going</span> <span m="2828050">to</span> <span m="2828110">let</span> <span m="2828270">R</span> <span m="2828680">be</span> <span m="2828910">that</span> <span m="2829130">random</span> <span m="2829440">variable.</span> <span m="2831720">It</span> <span m="2831850">will</span> <span m="2831950">be</span> <span m="2832120">the</span> <span m="2832680">step</span> <span m="2834230">when</span> <span m="2834470">failure</span> <span m="2835080">occurs,</span> <span m="2836300">first</span> <span m="2836640">failure.</span> </p>
<p><span m="2845120">We</span> <span m="2845200">want</span> <span m="2845480">to</span> <span m="2845540">know</span> <span m="2845700">what's</span> <span m="2845900">the</span> <span m="2846000">expected</span> <span m="2846520">value</span> <span m="2846870">of</span> <span m="2846980">R.</span> <span m="2848220">Mean</span> <span m="2848470">time</span> <span m="2848770">to</span> <span m="2848840">failure.</span> <span m="2852430">And</span> <span m="2852650">we're</span> <span m="2852730">going</span> <span m="2852860">to</span> <span m="2852960">use</span> <span m="2853330">that</span> <span m="2853640">top</span> <span m="2853950">formula</span> <span m="2854430">up</span> <span m="2854680">there.</span> <span m="2856350">Makes</span> <span m="2856580">it</span> <span m="2856640">a</span> <span m="2856680">lot</span> <span m="2856900">simpler</span> <span m="2858620">to</span> <span m="2858750">do</span> <span m="2858930">that</span> <span m="2859160">than</span> <span m="2859260">the</span> <span m="2859340">other</span> <span m="2859480">definitions.</span> <span m="2865010">Now,</span> <span m="2865830">the</span> <span m="2865950">probability</span> <span m="2866680">that</span> <span m="2866880">R</span> <span m="2867260">is</span> <span m="2867470">bigger</span> <span m="2867890">than</span> <span m="2868090">i</span> <span m="2870130">is</span> <span m="2870680">the</span> <span m="2870780">same</span> <span m="2871390">as</span> <span m="2872150">the</span> <span m="2872250">probability</span> <span m="2872910">it</span> <span m="2872990">did</span> <span m="2873130">not</span> <span m="2873520">fail</span> <span m="2874270">in</span> <span m="2874380">the</span> <span m="2874450">first</span> <span m="2874750">i</span> <span m="2874950">steps.</span> <span m="2878970">So</span> <span m="2879200">this</span> <span m="2880680">equals</span> <span m="2881310">the</span> <span m="2881440">probability</span> <span m="2882880">of</span> <span m="2883080">no</span> <span m="2883370">failure</span> <span m="2887450">in</span> <span m="2887770">the</span> <span m="2888090">first</span> <span m="2888500">i</span> <span m="2888670">steps.</span> </p>
<p><span m="2893900">Because</span> <span m="2894430">this</span> <span m="2894660">event,</span> <span m="2895030">R</span> <span m="2895260">bigger</span> <span m="2895480">than</span> <span m="2895620">i</span> <span m="2895840">means</span> <span m="2896300">that</span> <span m="2896450">the</span> <span m="2897400">first</span> <span m="2897760">failure</span> <span m="2898230">was</span> <span m="2898510">not</span> <span m="2898850">in</span> <span m="2898930">the</span> <span m="2898990">first</span> <span m="2899460">steps,</span> <span m="2899840">so</span> <span m="2899910">the</span> <span m="2899980">system</span> <span m="2900320">was</span> <span m="2900450">fine</span> <span m="2900910">for</span> <span m="2901010">the</span> <span m="2901070">first i</span> <span m="2901490">steps.</span> <span m="2903380">And</span> <span m="2903580">because</span> <span m="2903900">of</span> <span m="2904000">mutual</span> <span m="2904470">independence</span> <span m="2905970">on</span> <span m="2906170">when</span> <span m="2906390">failure</span> <span m="2906860">occurs,</span> <span m="2908380">this</span> <span m="2908680">is</span> <span m="2908780">simply</span> <span m="2909110">the</span> <span m="2909190">product</span> <span m="2910300">that</span> <span m="2911360">we're</span> <span m="2911640">OK,</span> <span m="2912470">no</span> <span m="2912700">failure</span> <span m="2913710">in</span> <span m="2913830">the</span> <span m="2913890">first</span> <span m="2914210">step</span> <span m="2917140">times</span> <span m="2917640">the</span> <span m="2917700">probability.</span> <span m="2918600">We're</span> <span m="2918810">OK</span> <span m="2920000">in</span> <span m="2920140">the</span> <span m="2920200">second</span> <span m="2920540">step</span> <span m="2924390">and</span> <span m="2924750">so</span> <span m="2924970">forth</span> <span m="2926450">up</span> <span m="2926780">to</span> <span m="2926940">the</span> <span m="2927100">ith</span> <span m="2927350">step.</span> <span m="2927800">We're OK</span> <span m="2928300">in</span> <span m="2928410">ith</span> <span m="2928680">step.</span> </p>
<p><span m="2932830">And</span> <span m="2933040">this</span> <span m="2933230">is</span> <span m="2933340">where</span> <span m="2933530">we're</span> <span m="2933610">using</span> <span m="2933970">mutual</span> <span m="2934300">independence,</span> <span m="2935040">because</span> <span m="2935210">I've</span> <span m="2935340">gone</span> <span m="2935650">from</span> <span m="2935750">a</span> <span m="2935790">situation</span> <span m="2936400">where</span> <span m="2937420">we're OK</span> <span m="2937930">in</span> <span m="2938010">the</span> <span m="2938070">first</span> <span m="2938390">i</span> <span m="2938520">steps.</span> <span m="2938990">The</span> <span m="2939060">probability</span> <span m="2939500">of</span> <span m="2939580">that</span> <span m="2940420">is</span> <span m="2940600">the</span> <span m="2940680">product</span> <span m="2941210">of</span> <span m="2941270">the</span> <span m="2941330">probabilities.</span> <span m="2942280">We're</span> <span m="2942480">OK</span> <span m="2942820">in</span> <span m="2942880">each</span> <span m="2943030">step</span> <span m="2943280">individually.</span> <span m="2944860">So</span> <span m="2945020">that</span> <span m="2945270">needed</span> <span m="2945760">independents.</span> </p>
<p><span m="2949890">Well,</span> <span m="2951220">what</span> <span m="2951470">is</span> <span m="2951670">the</span> <span m="2951760">probability</span> <span m="2952260">we're</span> <span m="2952390">OK</span> <span m="2952670">in</span> <span m="2952750">the</span> <span m="2952820">first</span> <span m="2953070">step?</span> <span m="2957610">What's</span> <span m="2957730">the</span> <span m="2957820">probability</span> <span m="2958230">of</span> <span m="2958320">no</span> <span m="2958600">failure</span> <span m="2959110">on</span> <span m="2959220">step</span> <span m="2959520">one?</span> <span m="2961208">1</span> <span m="2961630">minus</span> <span m="2961990">p,</span> <span m="2962370">because</span> <span m="2962540">p</span> <span m="2962860">is</span> <span m="2963010">the probability</span> <span m="2963340">we</span> <span m="2963710">did</span> <span m="2963960">fail.</span> <span m="2964950">So</span> <span m="2965160">we're</span> <span m="2965280">OK</span> <span m="2965610">with</span> <span m="2965750">probability</span> <span m="2966110">1</span> <span m="2966300">minus</span> <span m="2966600">p.</span> <span m="2968120">What</span> <span m="2968240">is</span> <span m="2968300">the</span> <span m="2968370">probability</span> <span m="2968860">we're</span> <span m="2969000">OK</span> <span m="2969290">in</span> <span m="2969380">the</span> <span m="2969440">second</span> <span m="2969800">step?</span> <span m="2971930">1</span> <span m="2972150">minus</span> <span m="2972460">p</span> <span m="2974020">and</span> <span m="2974370">so</span> <span m="2974670">forth</span> <span m="2976580">for</span> <span m="2976770">all</span> <span m="2977090">i</span> <span m="2977280">steps.</span> <span m="2979040">So</span> <span m="2979190">this</span> <span m="2979360">is</span> <span m="2979470">1</span> <span m="2979660">minus</span> <span m="2979970">p</span> <span m="2980190">to</span> <span m="2980250">the</span> <span m="2980370">eye.</span> </p>
<p><span m="2982480">And</span> <span m="2982880">it's</span> <span m="2983010">usually</span> <span m="2983370">simpler</span> <span m="2983720">to</span> <span m="2983820">write</span> <span m="2984150">that</span> <span m="2985170">as</span> <span m="2986400">alpha</span> <span m="2986840">to the</span> <span m="2986940">i</span> <span m="2988960">where</span> <span m="2989840">alpha</span> <span m="2990400">equals</span> <span m="2990810">1</span> <span m="2990990">minus</span> <span m="2991320">p.</span> <span m="2993200">Because</span> <span m="2993630">what</span> <span m="2993720">we've</span> <span m="2993870">got</span> <span m="2994160">here</span> <span m="2994350">now</span> <span m="2996970">is</span> <span m="2997270">the</span> <span m="2997360">sum.</span> <span m="2999150">Expected</span> <span m="2999550">value</span> <span m="2999810">of</span> <span m="2999910">R is</span> <span m="3000190">a</span> <span m="3000250">sum,</span> <span m="3001110">i</span> <span m="3001460">equals</span> <span m="3001890">0</span> <span m="3002360">to</span> <span m="3002520">infinity</span> <span m="3004580">of</span> <span m="3005470">that</span> <span m="3005690">probability,</span> <span m="3007850">which</span> <span m="3008100">is</span> <span m="3008200">just</span> <span m="3008510">alpha</span> <span m="3008810">to</span> <span m="3008870">the</span> <span m="3009000">i.</span> <span m="3010990">And</span> <span m="3011140">we</span> <span m="3011250">all</span> <span m="3011490">know</span> <span m="3011610">what</span> <span m="3011750">that</span> <span m="3011950">sum</span> <span m="3012200">is,</span> <span m="3012400">right?</span> <span m="3014280">What's</span> <span m="3014470">that</span> <span m="3014650">sum?</span> <span m="3019430">1</span> <span m="3019830">over</span> <span m="3020090">1</span> <span m="3020340">minus</span> <span m="3020710">alpha.</span> </p>
<p><span m="3023640">And</span> <span m="3023830">that</span> <span m="3024260">plug</span> <span m="3024510">back</span> <span m="3024750">in</span> <span m="3024870">the</span> <span m="3024990">alpha</span> <span m="3026030">to</span> <span m="3026160">be</span> <span m="3026270">1</span> <span m="3026470">minus</span> <span m="3026790">p</span> <span m="3028340">is</span> <span m="3029120">1</span> <span m="3029466">over</span> <span m="3029812">1</span> <span m="3030160">minus</span> <span m="3030720">1</span> <span m="3030930">minus</span> <span m="3031260">p.</span> <span m="3032810">And</span> <span m="3032950">that's</span> <span m="3033150">very</span> <span m="3033350">simple.</span> <span m="3033660">That's</span> <span m="3033880">1</span> <span m="3034260">over</span> <span m="3034510">p.</span> <span m="3037300">So</span> <span m="3037390">the</span> <span m="3037500">expected</span> <span m="3038170">time</span> <span m="3038520">to</span> <span m="3038590">fail,</span> <span m="3039080">the</span> <span m="3039440">expected</span> <span m="3039880">step</span> <span m="3040310">of</span> <span m="3040750">when</span> <span m="3041010">you're</span> <span m="3041100">going</span> <span m="3041220">to</span> <span m="3041280">fail,</span> <span m="3041580">is</span> <span m="3041690">1</span> <span m="3042000">over</span> <span m="3042180">p</span> <span m="3042710">where</span> <span m="3043210">p</span> <span m="3043480">is</span> <span m="3043600">the</span> <span m="3043680">failure</span> <span m="3044020">probability.</span> </p>
<p><span m="3045500">So</span> <span m="3045690">for</span> <span m="3045800">example,</span> <span m="3046280">if</span> <span m="3046360">you</span> <span m="3046420">have</span> <span m="3046510">a</span> <span m="3046560">1%</span> <span m="3047260">chance</span> <span m="3048140">of</span> <span m="3048280">failing</span> <span m="3048710">on</span> <span m="3048830">any</span> <span m="3048980">step,</span> <span m="3050120">your</span> <span m="3050290">mean</span> <span m="3050560">time</span> <span m="3050820">to</span> <span m="3050900">failure</span> <span m="3051980">is</span> <span m="3052170">what?</span> <span m="3054510">100.</span> <span m="3055322">1</span> <span m="3055730">over</span> <span m="3055960">0.01.</span> <span m="3057260">So</span> <span m="3057450">very</span> <span m="3057830">easy</span> <span m="3058110">to</span> <span m="3058170">compute</span> <span m="3058540">mean</span> <span m="3058730">time</span> <span m="3058950">to</span> <span m="3059030">failure.</span> <span m="3061870">Any</span> <span m="3062070">questions</span> <span m="3063980">about</span> <span m="3064240">that?</span> <span m="3066850">Of</span> <span m="3066950">course,</span> <span m="3067110">you</span> <span m="3067210">can</span> <span m="3067250">do</span> <span m="3067420">it</span> <span m="3067490">with</span> <span m="3067580">the</span> <span m="3067660">other</span> <span m="3067820">definitions,</span> <span m="3068430">but</span> <span m="3068540">the</span> <span m="3068610">calculations</span> <span m="3069390">are</span> <span m="3069480">a</span> <span m="3069520">little</span> <span m="3070420">more</span> <span m="3070590">painful.</span> <span m="3071660">Yeah.</span> </p>
<p><span m="3073220">AUDIENCE: Why are we</span> <span m="3073563">summing it?</span> <span m="3073906">Is it</span> <span m="3074250">like</span> <span m="3074750">an</span> <span m="3075000">accumulative</span> <span m="3075750">solution</span> <span m="3076250">basically?</span> </p>
<p><span m="3077250">PROFESSOR: Why</span> <span m="3077710">are we</span> <span m="3077990">summing?</span> <span m="3078860">Well,</span> <span m="3079660">that's</span> <span m="3079870">the</span> <span m="3079940">definition.</span> <span m="3080830">The</span> <span m="3080890">theorem</span> <span m="3081290">says</span> <span m="3082980">the</span> <span m="3083090">expected</span> <span m="3083620">value</span> <span m="3083910">of a</span> <span m="3083970">random</span> <span m="3084400">variable</span> <span m="3085930">is</span> <span m="3086100">the</span> <span m="3086200">sum</span> <span m="3086960">of</span> <span m="3087050">the</span> <span m="3087110">probability</span> <span m="3087540">that</span> <span m="3088490">it's</span> <span m="3088730">bigger</span> <span m="3089000">than</span> <span m="3089130">i.</span> <span m="3090980">That's</span> <span m="3091150">what</span> <span m="3091240">the</span> <span m="3091310">theorem</span> <span m="3091690">says.</span> <span m="3092040">So we're</span> <span m="3092130">just</span> <span m="3092330">plugging</span> <span m="3092750">into</span> <span m="3092910">the</span> <span m="3093000">theorem.</span> <span m="3094540">And</span> <span m="3094660">then</span> <span m="3094760">the</span> <span m="3094820">theorem</span> <span m="3095170">was</span> <span m="3095320">proved</span> <span m="3097060">based</span> <span m="3097420">on</span> <span m="3098660">the</span> <span m="3098770">corollary,</span> <span m="3099450">which</span> <span m="3099640">came</span> <span m="3099950">from</span> <span m="3101440">the</span> <span m="3101530">theorem</span> <span m="3101860">and then</span> <span m="3102020">the</span> <span m="3102080">definition.</span> </p>
<p><span m="3103230">So</span> <span m="3103360">we</span> <span m="3103390">went</span> <span m="3103560">through</span> <span m="3103660">a</span> <span m="3103710">series</span> <span m="3104270">of</span> <span m="3104350">steps.</span> <span m="3104670">We</span> <span m="3104750">started</span> <span m="3104990">with</span> <span m="3105100">a</span> <span m="3105140">definition</span> <span m="3105810">of</span> <span m="3105870">expected</span> <span m="3106310">value,</span> <span m="3107220">which</span> <span m="3107510">makes</span> <span m="3107750">sense.</span> <span m="3108790">Then</span> <span m="3108930">we</span> <span m="3109010">got</span> <span m="3109150">another</span> <span m="3109430">way</span> <span m="3109560">to</span> <span m="3109630">compute</span> <span m="3110030">it</span> <span m="3110140">based</span> <span m="3110370">on</span> <span m="3110440">that</span> <span m="3110520">definition.</span> <span m="3113770">And</span> <span m="3113930">then</span> <span m="3114910">a</span> <span m="3115010">corollary</span> <span m="3115630">to</span> <span m="3115710">that</span> <span m="3116660">and</span> <span m="3116820">then</span> <span m="3116960">we</span> <span m="3117050">use</span> <span m="3117210">the</span> <span m="3117280">corollary</span> <span m="3117750">to</span> <span m="3117820">prove</span> <span m="3118040">another</span> <span m="3118330">way</span> <span m="3118490">of</span> <span m="3118560">computing</span> <span m="3118920">expected</span> <span m="3119330">values.</span> <span m="3119710">We</span> <span m="3119810">went</span> <span m="3120010">through</span> <span m="3120120">a</span> <span m="3120190">bunch</span> <span m="3120470">of</span> <span m="3120550">general</span> <span m="3120950">steps</span> <span m="3121830">and</span> <span m="3121930">then</span> <span m="3122050">we</span> <span m="3122160">used,</span> <span m="3122760">basically</span> <span m="3123130">you</span> <span m="3123200">could</span> <span m="3123310">use</span> <span m="3123520">this</span> <span m="3123670">as</span> <span m="3123780">a</span> <span m="3123830">definition</span> <span m="3124340">of</span> <span m="3124420">this</span> <span m="3124560">point</span> <span m="3125310">for</span> <span m="3125430">expected</span> <span m="3125840">value.</span> <span m="3126510">And</span> <span m="3126710">it just</span> <span m="3126930">says</span> <span m="3127070">you</span> <span m="3127160">sum</span> <span m="3127410">those</span> <span m="3127600">things</span> <span m="3127850">up</span> <span m="3128020">and</span> <span m="3128110">you</span> <span m="3128170">get</span> <span m="3128310">the</span> <span m="3128440">answer.</span> <span m="3132510">Any</span> <span m="3132820">other</span> <span m="3133130">questions?</span> </p>
<p><span m="3136990">OK,</span> <span m="3137340">there's</span> <span m="3137510">a</span> <span m="3137940">variation</span> <span m="3138670">on</span> <span m="3138760">this</span> <span m="3138920">problem</span> <span m="3139380">that</span> <span m="3139480">you</span> <span m="3139560">see</span> <span m="3139740">all</span> <span m="3139870">the</span> <span m="3139960">time</span> <span m="3140440">in</span> <span m="3141710">sort</span> <span m="3141910">of</span> <span m="3141980">trick</span> <span m="3142210">questions</span> <span m="3142890">or</span> <span m="3142970">in</span> <span m="3143050">the</span> <span m="3143280">popular</span> <span m="3143740">press</span> <span m="3144090">sometime,</span> <span m="3145440">that</span> <span m="3145650">often</span> <span m="3146410">confuses</span> <span m="3146940">people.</span> <span m="3147220">People</span> <span m="3147510">sometimes</span> <span m="3147950">think</span> <span m="3148130">of</span> <span m="3148200">it</span> <span m="3148260">as</span> <span m="3148370">a</span> <span m="3148430">paradox,</span> <span m="3149630">though</span> <span m="3149730">it's</span> <span m="3149740">not</span> <span m="3150030">really</span> <span m="3150390">one.</span> <span m="3151930">And the</span> <span m="3152070">example's</span> <span m="3152610">something</span> <span m="3152760">like</span> <span m="3152900">the</span> <span m="3152960">following.</span> </p>
<p><span m="3154060">Say</span> <span m="3154360">that</span> <span m="3154470">a</span> <span m="3154510">couple,</span> <span m="3155640">they're</span> <span m="3155950">going</span> <span m="3156080">to</span> <span m="3156150">have</span> <span m="3156350">kids.</span> <span m="3156610">What</span> <span m="3156710">they</span> <span m="3156820">really</span> <span m="3157110">want</span> <span m="3157380">is</span> <span m="3157470">a</span> <span m="3157520">baby</span> <span m="3157820">girl.</span> <span m="3158990">They</span> <span m="3159120">get</span> <span m="3159230">a</span> <span m="3159280">boy, fine,</span> <span m="3159920">but</span> <span m="3160010">that's</span> <span m="3160170">not</span> <span m="3160390">what</span> <span m="3160500">they're</span> <span m="3160590">concerned</span> <span m="3160990">about.</span> <span m="3161230">They</span> <span m="3161330">want</span> <span m="3161650">to</span> <span m="3161710">have</span> <span m="3161870">a</span> <span m="3161950">baby</span> <span m="3162220">girl.</span> <span m="3163440">And</span> <span m="3163930">let's</span> <span m="3164120">say</span> <span m="3164280">that</span> <span m="3164380">each</span> <span m="3164670">time</span> <span m="3164970">they</span> <span m="3165100">have</span> <span m="3165310">a</span> <span m="3165350">kid</span> <span m="3165670">it's</span> <span m="3165840">50-50</span> <span m="3166490">boy</span> <span m="3166750">or</span> <span m="3166840">girl.</span> <span m="3168300">And</span> <span m="3168820">let's</span> <span m="3169010">say</span> <span m="3169440">that</span> <span m="3169560">it's</span> <span m="3169720">mutually</span> <span m="3170190">independent</span> <span m="3171290">from</span> <span m="3171510">one</span> <span m="3171710">kid</span> <span m="3171890">to</span> <span m="3171960">the</span> <span m="3172020">next,</span> <span m="3172340">which</span> <span m="3172550">is</span> <span m="3172680">not</span> <span m="3172930">true</span> <span m="3173150">in</span> <span m="3173250">practice.</span> <span m="3174380">There</span> <span m="3174590">tends</span> <span m="3174860">to</span> <span m="3174950">be</span> <span m="3175260">correlation.</span> <span m="3175860">But</span> <span m="3175960">let's</span> <span m="3176180">assume</span> <span m="3176660">it's</span> <span m="3176800">mutually</span> <span m="3177170">independent</span> <span m="3177590">from</span> <span m="3177650">one</span> <span m="3177860">kid</span> <span m="3178060">to</span> <span m="3178130">the</span> <span m="3178190">next.</span> </p>
<p><span m="3179500">Now,</span> <span m="3180420">if</span> <span m="3180590">on</span> <span m="3180710">the</span> <span m="3180770">first</span> <span m="3181140">try</span> <span m="3181590">a</span> <span m="3181690">couple</span> <span m="3182030">get</span> <span m="3182200">a</span> <span m="3182250">girl,</span> <span m="3183750">great,</span> <span m="3184190">they're</span> <span m="3184390">done,</span> <span m="3184800">they</span> <span m="3184900">have</span> <span m="3185020">one</span> <span m="3185230">kid,</span> <span m="3185460">that's</span> <span m="3185620">it,</span> <span m="3186110">because</span> <span m="3186300">they</span> <span m="3186360">just</span> <span m="3186650">wanted</span> <span m="3186870">a</span> <span m="3186920">girl.</span> <span m="3187680">If</span> <span m="3187890">they</span> <span m="3187960">get</span> <span m="3188100">a</span> <span m="3188140">boy,</span> <span m="3188770">OK,</span> <span m="3189430">try</span> <span m="3189690">again.</span> <span m="3191140">And</span> <span m="3191310">they</span> <span m="3191400">keep</span> <span m="3191640">trying</span> <span m="3192070">again</span> <span m="3192570">until</span> <span m="3192980">they</span> <span m="3193100">get</span> <span m="3193540">the</span> <span m="3193650">girl,</span> <span m="3194890">even</span> <span m="3195130">if</span> <span m="3195220">it's</span> <span m="3195570">50</span> <span m="3195920">kids.</span> <span m="3196480">They</span> <span m="3196770">wait</span> <span m="3196990">till</span> <span m="3197090">they</span> <span m="3197160">get</span> <span m="3197290">the</span> <span m="3197370">girl.</span> <span m="3198570">And</span> <span m="3198740">the</span> <span m="3198800">question</span> <span m="3199260">is,</span> <span m="3200050">how</span> <span m="3200310">many</span> <span m="3201030">baby</span> <span m="3201350">boys</span> <span m="3201830">do</span> <span m="3201910">you</span> <span m="3201990">expect</span> <span m="3202430">to</span> <span m="3202510">get</span> <span m="3203800">before</span> <span m="3204190">you</span> <span m="3204310">have</span> <span m="3204910">the</span> <span m="3205020">girl?</span> <span m="3207450">what's?</span> <span m="3207690">The</span> <span m="3207780">expected</span> <span m="3208290">number</span> <span m="3208540">of</span> <span m="3208630">boys</span> <span m="3209050">to</span> <span m="3209160">get</span> <span m="3209450">the</span> <span m="3209580">girl,</span> <span m="3210380">then</span> <span m="3210540">you</span> <span m="3210620">quit?</span> <span m="3212461">So</span> <span m="3212948">let's</span> <span m="3216357">do</span> <span m="3216844">that.</span> </p>
<p><span m="3228070">So</span> <span m="3228420">we</span> <span m="3228630">want</span> <span m="3228850">to</span> <span m="3228920">know.</span> <span m="3229735">We</span> <span m="3230250">have</span> <span m="3230410">the</span> <span m="3230480">following</span> <span m="3231030">data.</span> <span m="3232690">The</span> <span m="3232780">probability</span> <span m="3233360">of</span> <span m="3233430">a</span> <span m="3233500">boy</span> <span m="3235210">is</span> <span m="3235520">1/2.</span> <span m="3237860">You</span> <span m="3237990">keep</span> <span m="3238310">having</span> <span m="3238600">boys</span> <span m="3239410">until</span> <span m="3239630">you</span> <span m="3239710">get</span> <span m="3239840">a</span> <span m="3239880">girl,</span> <span m="3240160">then</span> <span m="3240290">you</span> <span m="3240370">quit.</span> <span m="3246840">We're</span> <span m="3246970">going</span> <span m="3247100">to</span> <span m="3247160">let</span> <span m="3247460">R</span> <span m="3247910">be</span> <span m="3248440">the</span> <span m="3248580">random</span> <span m="3250320">variable</span> <span m="3250780">for</span> <span m="3250900">the</span> <span m="3250980">number</span> <span m="3251160">of</span> <span m="3251230">boys.</span> <span m="3254050">And</span> <span m="3254900">everything</span> <span m="3255280">is</span> <span m="3255420">mutually</span> <span m="3255840">independent</span> <span m="3260480">from</span> <span m="3260790">one</span> <span m="3260960">child</span> <span m="3261290">to</span> <span m="3261370">the</span> <span m="3261440">next.</span> </p>
<p><span m="3263300">How</span> <span m="3263490">many</span> <span m="3263660">people</span> <span m="3263960">think</span> <span m="3264780">you</span> <span m="3264960">expect</span> <span m="3265500">to</span> <span m="3265610">have</span> <span m="3265960">more</span> <span m="3266350">boys</span> <span m="3267910">than</span> <span m="3268110">the</span> <span m="3268180">one</span> <span m="3268430">girl?</span> <span m="3270330">You</span> <span m="3270440">keep</span> <span m="3270670">having</span> <span m="3270890">boys</span> <span m="3271150">till</span> <span m="3271190">you</span> <span m="3271230">get</span> <span m="3271360">the</span> <span m="3271430">girl.</span> <span m="3272680">Most</span> <span m="3273130">people</span> <span m="3273370">think</span> <span m="3273660">this.</span> <span m="3273880">How</span> <span m="3274020">many</span> <span m="3274170">people</span> <span m="3274430">think</span> <span m="3275130">you</span> <span m="3275280">expect</span> <span m="3275680">to</span> <span m="3275760">have</span> <span m="3275950">fewer</span> <span m="3276410">than</span> <span m="3276570">one</span> <span m="3276820">boy?</span> <span m="3279390">Nobody.</span> <span m="3280630">How</span> <span m="3280720">many</span> <span m="3280870">people</span> <span m="3281120">think</span> <span m="3281370">you</span> <span m="3281480">expect</span> <span m="3281820">to</span> <span m="3281910">have</span> <span m="3282050">an</span> <span m="3282120">equal</span> <span m="3282370">number?</span> <span m="3283136">Expect</span> <span m="3283800">one</span> <span m="3283980">boy.</span> <span m="3284160">Good,</span> <span m="3284480">OK,</span> <span m="3284800">so</span> <span m="3284910">that's</span> <span m="3285120">the</span> <span m="3285270">answer.</span> <span m="3286140">In</span> <span m="3286170">fact,</span> <span m="3286420">you</span> <span m="3286500">expect</span> <span m="3287020">to</span> <span m="3287120">have</span> <span m="3287330">just</span> <span m="3287550">one</span> <span m="3287760">boy.</span> <span m="3288700">And</span> <span m="3289020">the</span> <span m="3289090">proof,</span> <span m="3290010">we</span> <span m="3290260">sort</span> <span m="3290410">of</span> <span m="3290480">just</span> <span m="3290780">did</span> <span m="3291010">it</span> <span m="3291110">here.</span> </p>
<p><span m="3293100">Now,</span> <span m="3293390">in</span> <span m="3293550">this</span> <span m="3293770">case,</span> <span m="3294160">we're</span> <span m="3294260">going</span> <span m="3294360">to</span> <span m="3294460">set it</span> <span m="3294550">up</span> <span m="3294650">as</span> <span m="3294770">a</span> <span m="3294810">mean</span> <span m="3295070">time</span> <span m="3295350">to</span> <span m="3295430">failure.</span> <span m="3296460">Same</span> <span m="3296800">kind</span> <span m="3297000">of</span> <span m="3297080">thing.</span> <span m="3298480">Now,</span> <span m="3299120">in</span> <span m="3299260">this</span> <span m="3299430">case</span> <span m="3299720">the</span> <span m="3299800">failure</span> <span m="3300270">mode,</span> <span m="3301170">they</span> <span m="3301270">want</span> <span m="3301570">the</span> <span m="3301660">girl,</span> <span m="3301950">but</span> <span m="3302050">that's</span> <span m="3302220">when</span> <span m="3302340">you</span> <span m="3302410">stop,</span> <span m="3302890">so</span> <span m="3303010">that's</span> <span m="3303230">the</span> <span m="3303310">failure</span> <span m="3303680">mode.</span> <span m="3305640">And</span> <span m="3305880">a</span> <span m="3305910">working</span> <span m="3306290">step</span> <span m="3306600">is</span> <span m="3306680">you</span> <span m="3306750">have</span> <span m="3306860">a</span> <span m="3306920">boy</span> <span m="3307210">and</span> <span m="3307300">you</span> <span m="3307360">keep</span> <span m="3307520">having</span> <span m="3307700">the</span> <span m="3307760">working</span> <span m="3308080">steps</span> <span m="3308400">until</span> <span m="3308600">you</span> <span m="3308720">have</span> <span m="3308860">failure</span> <span m="3309220">mode</span> <span m="3309440">and</span> <span m="3309530">then</span> <span m="3309640">you</span> <span m="3309720">stop.</span> <span m="3310570">And</span> <span m="3310720">in</span> <span m="3310780">this</span> <span m="3310930">case,</span> <span m="3311160">we're</span> <span m="3311270">not</span> <span m="3312140">counting</span> <span m="3312630">the</span> <span m="3312690">step</span> <span m="3312980">when</span> <span m="3313090">you</span> <span m="3313160">stop.</span> </p>
<p><span m="3313610">So</span> <span m="3314410">we</span> <span m="3314680">know</span> <span m="3315040">from</span> <span m="3315270">that</span> <span m="3316082">that</span> <span m="3316490">the</span> <span m="3316580">expected</span> <span m="3317150">value</span> <span m="3317520">of</span> <span m="3317630">R</span> <span m="3319270">is</span> <span m="3320100">1</span> <span m="3320360">over</span> <span m="3320560">p,</span> <span m="3321150">which</span> <span m="3321290">is</span> <span m="3321360">going</span> <span m="3321480">to</span> <span m="3321540">be</span> <span m="3321640">1/2.</span> <span m="3322620">That's</span> <span m="3322910">the</span> <span m="3322990">number</span> <span m="3323360">of</span> <span m="3323440">children</span> <span m="3323880">you</span> <span m="3323980">have.</span> <span m="3326420">And</span> <span m="3326560">that</span> <span m="3326740">equals</span> <span m="3327040">the</span> <span m="3327110">mean</span> <span m="3327440">time</span> <span m="3327770">to</span> <span m="3327850">failure.</span> <span m="3330170">Minus</span> <span m="3331420">the</span> <span m="3331560">girl.</span> <span m="3334250">Because</span> <span m="3334380">you</span> <span m="3334670">count</span> <span m="3334930">the</span> <span m="3335000">girl</span> <span m="3335330">as</span> <span m="3335420">one</span> <span m="3335640">of</span> <span m="3335720">the</span> <span m="3335790">children.</span> <span m="3337810">And</span> <span m="3337990">that's</span> <span m="3338230">going</span> <span m="3338350">to</span> <span m="3338420">be</span> <span m="3338590">the</span> <span m="3338680">expected</span> <span m="3339130">number</span> <span m="3339360">of</span> <span m="3339430">boys.</span> <span m="3340280">So</span> <span m="3340420">number</span> <span m="3340670">of</span> <span m="3340740">children</span> <span m="3341180">you're</span> <span m="3341240">expected</span> <span m="3341580">to</span> <span m="3341660">have</span> <span m="3341890">minus</span> <span m="3342570">the</span> <span m="3342690">girl.</span> <span m="3343880">This</span> <span m="3344120">is</span> <span m="3344230">1</span> <span m="3344480">over</span> <span m="3344720">1/2.</span> <span m="3349115">And</span> <span m="3349600">that</span> <span m="3349790">is</span> <span m="3349960">2</span> <span m="3350110">minus</span> <span m="3350440">1</span> <span m="3350710">equals</span> <span m="3351050">1.</span> <span m="3353330">So you</span> <span m="3353700">expect</span> <span m="3354120">to</span> <span m="3354220">have</span> <span m="3354420">one</span> <span m="3354700">boy</span> <span m="3356300">before</span> <span m="3356650">you</span> <span m="3356760">get</span> <span m="3356930">the</span> <span m="3357000">girl.</span> <span m="3358770">Any</span> <span m="3358940">questions</span> <span m="3361160">on</span> <span m="3361310">that?</span> </p>
<p><span m="3362930">OK,</span> <span m="3363670">how</span> <span m="3363840">about</span> <span m="3364070">this?</span> <span m="3366180">Some</span> <span m="3366480">couples</span> <span m="3366850">want</span> <span m="3367030">to</span> <span m="3367090">have</span> <span m="3367450">at</span> <span m="3367550">least</span> <span m="3367990">one</span> <span m="3368260">of</span> <span m="3368360">each</span> <span m="3368550">sex.</span> <span m="3369840">They</span> <span m="3369940">want</span> <span m="3370160">to</span> <span m="3370220">have</span> <span m="3370430">at</span> <span m="3370500">least</span> <span m="3370730">one</span> <span m="3370960">boy</span> <span m="3371320">and</span> <span m="3371450">one</span> <span m="3371680">girl.</span> <span m="3373530">So</span> <span m="3373730">they</span> <span m="3373870">keep</span> <span m="3374210">having</span> <span m="3374580">children</span> <span m="3375010">until</span> <span m="3375290">they</span> <span m="3375390">get</span> <span m="3375570">one</span> <span m="3375750">of</span> <span m="3375850">each</span> <span m="3376170">and</span> <span m="3376180">then</span> <span m="3376300">they</span> <span m="3376390">stop.</span> <span m="3378170">How</span> <span m="3378420">many</span> <span m="3378600">children</span> <span m="3379010">do</span> <span m="3379130">they</span> <span m="3379270">expect</span> <span m="3379620">to</span> <span m="3379710">have?</span> <span m="3385910">Somebody</span> <span m="3386220">said</span> <span m="3386560">two.</span> <span m="3387580">That's</span> <span m="3387770">a</span> <span m="3387820">minimum</span> <span m="3388340">number.</span> <span m="3390820">So</span> <span m="3391100">it's</span> <span m="3391270">not</span> <span m="3391490">likely</span> <span m="3391760">to</span> <span m="3391830">be</span> <span m="3391920">the</span> <span m="3392030">expected</span> <span m="3392630">number.</span> <span m="3394380">Three.</span> <span m="3395580">Well,</span> <span m="3395930">OK,</span> <span m="3396220">who</span> <span m="3396400">said</span> <span m="3396590">three?</span> <span m="3397730">Why</span> <span m="3397970">do</span> <span m="3398040">you</span> <span m="3398110">think</span> <span m="3398310">three?</span> </p>
<p><span m="3399030">AUDIENCE: Because</span> <span m="3400378">you have</span> <span m="3400872">to have</span> <span m="3401366">at least</span> <span m="3401860">the</span> <span m="3402354">first two,</span> <span m="3402848">so it</span> <span m="3403342">can be</span> <span m="3403836">greater</span> <span m="3404330">than</span> <span m="3404824">one.</span> <span m="3405820">I mean,</span> <span m="3406070">one</span> <span m="3406566">probability</span> <span m="3407062">is greater</span> <span m="3407558">than one</span> <span m="3408054">child, one</span> <span m="3408550">probability</span> <span m="3409046">is greater</span> <span m="3409542">than two</span> <span m="3410038">children.</span> <span m="3410534">And after that</span> <span m="3411030">it's</span> <span m="3412022">halves</span> <span m="3412518">which</span> <span m="3413014">add</span> <span m="3413510">up to</span> <span m="3414006">one.</span> </p>
<p><span m="3415700">PROFESSOR: Yeah,</span> <span m="3415840">that's</span> <span m="3416070">right.</span> <span m="3416330">That's a</span> <span m="3416510">good</span> <span m="3416620">proof.</span> <span m="3417080">Very</span> <span m="3417540">good.</span> <span m="3418000">That'll work.</span> <span m="3418850">Yeah,</span> <span m="3419290">yeah.</span> </p>
<p><span m="3420526">AUDIENCE: I have a question about</span> <span m="3420940">what</span> <span m="3421421">you were doing</span> <span m="3421902">before.</span> <span m="3422383">If</span> <span m="3422864">you</span> <span m="3423104">were</span> <span m="3423345">to switch</span> <span m="3423826">your</span> <span m="3424307">expectations</span> <span m="3424788">for a</span> <span m="3425269">girl and boy,</span> <span m="3425750">wouldn't it</span> <span m="3426231">come out</span> <span m="3426712">to the same</span> <span m="3427193">number</span> <span m="3427674">but</span> <span m="3428155">wouldn't it</span> <span m="3428636">kind</span> <span m="3428876">of</span> <span m="3429117">contradict</span> <span m="3429598">itself?</span> </p>
<p><span m="3430570">PROFESSOR: No,</span> <span m="3430930">if</span> <span m="3431050">you</span> <span m="3431240">stopped</span> <span m="3431630">as soon</span> <span m="3431720">as</span> <span m="3432000">you</span> <span m="3432110">had</span> <span m="3432250">a</span> <span m="3432310">boy,</span> <span m="3432940">you'd</span> <span m="3433030">expect</span> <span m="3433330">to</span> <span m="3433400">have</span> <span m="3433560">one</span> <span m="3433800">girl</span> <span m="3433980">before</span> <span m="3434270">you</span> <span m="3434340">got</span> <span m="3434520">a</span> <span m="3434570">boy.</span> <span m="3435780">Totally</span> <span m="3436220">symmetric.</span> <span m="3437410">In</span> <span m="3437520">this</span> <span m="3437650">case,</span> <span m="3437860">we're</span> <span m="3437950">stopping</span> <span m="3438390">when</span> <span m="3438500">we</span> <span m="3438590">get</span> <span m="3438740">the</span> <span m="3438810">girl,</span> <span m="3439140">so we</span> <span m="3439230">expect</span> <span m="3439570">to</span> <span m="3439660">have</span> <span m="3439870">one</span> <span m="3440140">boy.</span> <span m="3441560">You</span> <span m="3441660">might</span> <span m="3441840">have</span> <span m="3442020">none,</span> <span m="3442510">you</span> <span m="3442600">might</span> <span m="3442740">have</span> <span m="3442880">one,</span> <span m="3443260">you</span> <span m="3443330">might</span> <span m="3443490">have</span> <span m="3443650">two.</span> <span m="3444890">and</span> <span m="3445240">as</span> <span m="3445310">he</span> <span m="3445390">mentions,</span> <span m="3445730">if you</span> <span m="3445800">put</span> <span m="3445940">the</span> <span m="3446010">probabilities</span> <span m="3446560">in</span> <span m="3446640">there,</span> <span m="3446770">they</span> <span m="3446880">will</span> <span m="3447110">work</span> <span m="3447320">out</span> <span m="3447480">the</span> <span m="3447560">right</span> <span m="3447750">way,</span> <span m="3448000">but</span> <span m="3448120">we</span> <span m="3448310">just</span> <span m="3448530">did</span> <span m="3448620">it</span> <span m="3448720">simpler</span> <span m="3449090">by</span> <span m="3449210">the</span> <span m="3449300">mean</span> <span m="3449480">time</span> <span m="3449680">to</span> <span m="3449760">failure.</span> </p>
<p><span m="3452840">And</span> <span m="3453050">in</span> <span m="3453120">fact,</span> <span m="3453370">there's</span> <span m="3453540">also</span> <span m="3454330">another</span> <span m="3454590">proof,</span> <span m="3454910">what</span> <span m="3455360">you're</span> <span m="3455480">doing.</span> <span m="3457220">Another</span> <span m="3457400">way</span> <span m="3457510">to</span> <span m="3457580">think</span> <span m="3457830">about</span> <span m="3458340">how</span> <span m="3458620">many</span> <span m="3459270">kids</span> <span m="3459530">you</span> <span m="3459650">have</span> <span m="3459890">to</span> <span m="3459960">get</span> <span m="3460130">one</span> <span m="3460300">of</span> <span m="3460390">each.</span> <span m="3461400">Well you</span> <span m="3461590">have</span> <span m="3461810">the</span> <span m="3461870">first</span> <span m="3462170">kid.</span> <span m="3463880">And</span> <span m="3464130">now</span> <span m="3464750">you</span> <span m="3464890">have</span> <span m="3465700">this</span> <span m="3465920">problem,</span> <span m="3466960">because</span> <span m="3467160">you</span> <span m="3467260">want</span> <span m="3467600">one</span> <span m="3467780">of</span> <span m="3467840">the</span> <span m="3467940">other</span> <span m="3468090">sex.</span> <span m="3469200">And</span> <span m="3469360">you</span> <span m="3469430">keep</span> <span m="3469670">on</span> <span m="3469810">trying</span> <span m="3470110">to</span> <span m="3470190">get</span> <span m="3470340">one</span> <span m="3470490">of</span> <span m="3470550">the</span> <span m="3470630">other</span> <span m="3470810">sex</span> <span m="3471980">and</span> <span m="3472680">you</span> <span m="3472760">expect</span> <span m="3473130">to</span> <span m="3473230">have</span> <span m="3474220">two</span> <span m="3474510">children</span> <span m="3475850">to</span> <span m="3475950">get</span> <span m="3476120">one</span> <span m="3476310">of</span> <span m="3476370">the</span> <span m="3476460">other</span> <span m="3476620">sex.</span> </p>
<p><span m="3477090">So</span> <span m="3477210">you</span> <span m="3477270">have</span> <span m="3477370">one</span> <span m="3477680">to</span> <span m="3477740">start.</span> <span m="3478420">Whatever</span> <span m="3478710">it</span> <span m="3478770">is</span> <span m="3478910">doesn't</span> <span m="3479110">matter.</span> <span m="3479760">And</span> <span m="3479930">now</span> <span m="3480030">you</span> <span m="3480130">expect</span> <span m="3480400">to</span> <span m="3480480">have</span> <span m="3480640">two</span> <span m="3480820">more</span> <span m="3481690">until</span> <span m="3481910">you</span> <span m="3482040">hit</span> <span m="3482240">the</span> <span m="3482360">other</span> <span m="3482540">sex.</span> <span m="3483280">So</span> <span m="3483460">a</span> <span m="3483500">total</span> <span m="3483810">of</span> <span m="3483900">three</span> <span m="3484240">is</span> <span m="3484360">the</span> <span m="3484420">expected</span> <span m="3484820">number</span> <span m="3485020">of</span> <span m="3485080">children</span> <span m="3485970">to</span> <span m="3486080">get</span> <span m="3487230">one</span> <span m="3487470">boy</span> <span m="3487650">and</span> <span m="3487750">one</span> <span m="3487880">girl,</span> <span m="3488120">at</span> <span m="3488190">least.</span> <span m="3490985">Any</span> <span m="3491630">questions</span> <span m="3492690">about</span> <span m="3492870">expectation?</span> </p>
<p><span m="3495980">OK,</span> <span m="3497540">let's</span> <span m="3497720">do</span> <span m="3499150">another</span> <span m="3499550">example.</span> <span m="3502010">This</span> <span m="3502160">comes</span> <span m="3502390">up</span> <span m="3502680">all</span> <span m="3503040">the</span> <span m="3503140">time</span> <span m="3503650">in</span> <span m="3503720">experimental</span> <span m="3504390">work.</span> <span m="3506120">Probably</span> <span m="3506410">all</span> <span m="3506820">of</span> <span m="3506910">you</span> <span m="3507350">are</span> <span m="3507460">going</span> <span m="3507620">to</span> <span m="3507730">have</span> <span m="3508010">some</span> <span m="3508380">example</span> <span m="3508920">at</span> <span m="3509010">some</span> <span m="3509210">point</span> <span m="3509540">where</span> <span m="3509640">you're</span> <span m="3510270">going</span> <span m="3510410">to</span> <span m="3510490">do</span> <span m="3510600">a</span> <span m="3510670">problem</span> <span m="3511050">like</span> <span m="3511290">this.</span> <span m="3513050">And</span> <span m="3513280">most</span> <span m="3513530">all</span> <span m="3513650">the</span> <span m="3513740">time,</span> <span m="3514070">people</span> <span m="3514400">do</span> <span m="3514590">it</span> <span m="3514890">wrong.</span> <span m="3517020">So</span> <span m="3517500">let's</span> <span m="3517840">see</span> <span m="3518030">an</span> <span m="3518100">example.</span> </p>
<p><span m="3519520">Say</span> <span m="3519870">that</span> <span m="3520730">you</span> <span m="3520980">want</span> <span m="3521210">to</span> <span m="3521280">measure</span> <span m="3522510">the</span> <span m="3522640">latency</span> <span m="3523290">of</span> <span m="3523380">some</span> <span m="3523670">communications</span> <span m="3524390">channel.</span> <span m="3526410">And</span> <span m="3526600">you</span> <span m="3526680">want</span> <span m="3526850">to</span> <span m="3526910">know</span> <span m="3527010">what's</span> <span m="3527200">the</span> <span m="3527310">average</span> <span m="3527660">latency.</span> <span m="3529490">So</span> <span m="3529840">you</span> <span m="3530200">set</span> <span m="3530480">up</span> <span m="3530580">an</span> <span m="3530650">experiment.</span> <span m="3532570">You</span> <span m="3532910">send</span> <span m="3533170">a</span> <span m="3533210">package</span> <span m="3533600">to</span> <span m="3533680">the</span> <span m="3533750">channel,</span> <span m="3534180">you</span> <span m="3534310">measure</span> <span m="3534590">when</span> <span m="3534720">it</span> <span m="3534810">started,</span> <span m="3535250">when</span> <span m="3535360">it</span> <span m="3535420">got</span> <span m="3535570">to</span> <span m="3535660">the</span> <span m="3535750">other</span> <span m="3535980">end,</span> <span m="3536740">and</span> <span m="3536900">you</span> <span m="3536980">record</span> <span m="3537370">the</span> <span m="3537510">answer,</span> <span m="3538410">and</span> <span m="3538510">then</span> <span m="3538610">you</span> <span m="3538670">do</span> <span m="3538820">it</span> <span m="3538880">100</span> <span m="3539250">times</span> <span m="3540900">and</span> <span m="3541090">you</span> <span m="3541160">take</span> <span m="3541400">the</span> <span m="3541530">average.</span> <span m="3542940">And</span> <span m="3543090">you</span> <span m="3543160">say</span> <span m="3543350">that's</span> <span m="3543620">the</span> <span m="3543780">expected</span> <span m="3544270">latency</span> <span m="3544720">of</span> <span m="3544780">the</span> <span m="3544850">channel.</span> <span m="3546270">Very,</span> <span m="3546960">very</span> <span m="3547130">typical</span> <span m="3547660">kind</span> <span m="3547840">of</span> <span m="3547900">thing</span> <span m="3548040">to</span> <span m="3548100">do,</span> <span m="3548220">and</span> <span m="3548310">sometimes</span> <span m="3548770">OK</span> <span m="3549020">to</span> <span m="3549120">do.</span> </p>
<p><span m="3550840">So</span> <span m="3551040">you'd</span> <span m="3551160">do</span> <span m="3551320">something</span> <span m="3551500">like</span> <span m="3551720">this.</span> <span m="3552160">You</span> <span m="3554010">pick</span> <span m="3554180">a</span> <span m="3554250">random</span> <span m="3554570">variable</span> <span m="3554930">D,</span> <span m="3555510">which</span> <span m="3555690">is</span> <span m="3555790">going</span> <span m="3555920">to</span> <span m="3556000">denote</span> <span m="3556350">the</span> <span m="3556420">delay</span> <span m="3559400">of</span> <span m="3559630">a</span> <span m="3559700">packet</span> <span m="3563080">going</span> <span m="3563400">through</span> <span m="3563530">the</span> <span m="3563610">channel.</span> <span m="3568560">And</span> <span m="3568850">there's,</span> <span m="3569020">of</span> <span m="3569080">course,</span> <span m="3569360">some</span> <span m="3569720">underlying</span> <span m="3570360">distribution</span> <span m="3571010">here,</span> <span m="3571230">which</span> <span m="3571410">we'll</span> <span m="3571540">denote</span> <span m="3571920">by</span> <span m="3572750">f</span> <span m="3572940">of</span> <span m="3573080">x.</span> <span m="3575990">And</span> <span m="3576110">that's</span> <span m="3576300">just</span> <span m="3576560">the</span> <span m="3576640">probability</span> <span m="3577390">that</span> <span m="3577530">D</span> <span m="3577850">equals</span> <span m="3578250">x.</span> <span m="3578500">That's</span> <span m="3578660">the</span> <span m="3578730">probability</span> <span m="3579390">distribution</span> <span m="3580010">function.</span> </p>
<p><span m="3586450">And</span> <span m="3587980">as</span> <span m="3588090">part</span> <span m="3588280">of</span> <span m="3588360">doing</span> <span m="3588610">your</span> <span m="3588710">stuff,</span> <span m="3589010">you</span> <span m="3589120">notice</span> <span m="3589660">that</span> <span m="3590530">if</span> <span m="3590800">I</span> <span m="3590960">look</span> <span m="3591160">at</span> <span m="3591340">plot</span> <span m="3591820">f here,</span> <span m="3592110">if I</span> <span m="3592170">look</span> <span m="3592320">at</span> <span m="3592410">x</span> <span m="3592650">on</span> <span m="3592730">that</span> <span m="3592940">axis</span> <span m="3593550">and</span> <span m="3594240">f</span> <span m="3594470">of</span> <span m="3594600">x</span> <span m="3594810">here,</span> <span m="3598320">it</span> <span m="3598760">looks</span> <span m="3598980">something</span> <span m="3599480">like</span> <span m="3599750">this.</span> <span m="3602910">The</span> <span m="3603030">chance</span> <span m="3603470">of</span> <span m="3603570">getting</span> <span m="3603950">the</span> <span m="3604330">observed</span> <span m="3605100">cases</span> <span m="3605550">where</span> <span m="3605670">I</span> <span m="3605730">got</span> <span m="3605890">a</span> <span m="3605920">long</span> <span m="3606350">delay</span> <span m="3606690">were</span> <span m="3606820">small,</span> <span m="3607430">low</span> <span m="3607630">probability.</span> <span m="3609290">And</span> <span m="3609530">almost</span> <span m="3609840">all</span> <span m="3609960">the</span> <span m="3610040">time</span> <span m="3610280">I</span> <span m="3610320">got</span> <span m="3610460">a</span> <span m="3610520">short</span> <span m="3610790">delay.</span> <span m="3611890">So</span> <span m="3612100">very</span> <span m="3612510">typically</span> <span m="3612990">you'll</span> <span m="3613100">get</span> <span m="3613230">a</span> <span m="3613270">curve</span> <span m="3613740">that</span> <span m="3613870">looks</span> <span m="3614120">something</span> <span m="3614540">like</span> <span m="3614780">that</span> <span m="3615650">in</span> <span m="3615780">terms</span> <span m="3616060">of</span> <span m="3616150">the</span> <span m="3616660">observations</span> <span m="3617430">of</span> <span m="3617540">your</span> <span m="3617680">experiment.</span> </p>
<p><span m="3620600">And</span> <span m="3621010">say</span> <span m="3621900">that</span> <span m="3622020">you</span> <span m="3622110">do</span> <span m="3622250">your</span> <span m="3622400">experiment</span> <span m="3624250">100</span> <span m="3624750">times</span> <span m="3625930">and</span> <span m="3626090">the</span> <span m="3626300">average</span> <span m="3626900">latency</span> <span m="3627840">was</span> <span m="3628040">10</span> <span m="3628310">milliseconds.</span> <span m="3631940">And</span> <span m="3632050">sometimes</span> <span m="3632530">you</span> <span m="3632640">want</span> <span m="3632790">to</span> <span m="3632850">be</span> <span m="3632940">really</span> <span m="3633320">careful,</span> <span m="3633720">so</span> <span m="3633820">you</span> <span m="3633890">do</span> <span m="3634020">the</span> <span m="3634110">whole</span> <span m="3634350">thing</span> <span m="3634530">again,</span> <span m="3635050">you</span> <span m="3635140">do</span> <span m="3635270">another</span> <span m="3635590">100</span> <span m="3635920">times.</span> <span m="3637150">And</span> <span m="3637330">maybe</span> <span m="3637540">it's</span> <span m="3637690">nine</span> <span m="3638010">milliseconds</span> <span m="3638570">the</span> <span m="3638640">next</span> <span m="3638850">time.</span> <span m="3639190">And</span> <span m="3639310">that</span> <span m="3639440">sort</span> <span m="3639590">of</span> <span m="3639650">confirms</span> <span m="3640300">your</span> <span m="3640410">belief</span> <span m="3641280">that</span> <span m="3641430">your</span> <span m="3641550">first</span> <span m="3641890">experiment</span> <span m="3642320">was</span> <span m="3642450">valid</span> <span m="3643250">and</span> <span m="3643480">that</span> <span m="3643580">the</span> <span m="3643690">expected</span> <span m="3644280">latency</span> <span m="3644760">on</span> <span m="3644880">this</span> <span m="3645030">channel</span> <span m="3645430">is</span> <span m="3645540">10</span> <span m="3645760">milliseconds.</span> </p>
<p><span m="3648510">Sounds</span> <span m="3648870">pretty</span> <span m="3649060">good.</span> <span m="3651390">But</span> <span m="3651530">it</span> <span m="3651600">can</span> <span m="3651720">be</span> <span m="3651940">completely</span> <span m="3652560">wrong.</span> <span m="3653710">And</span> <span m="3653860">not</span> <span m="3654030">just</span> <span m="3654200">because</span> <span m="3654370">you're</span> <span m="3654490">unlucky,</span> <span m="3655390">but</span> <span m="3655560">because</span> <span m="3655910">you're</span> <span m="3656040">taking</span> <span m="3656550">a</span> <span m="3656650">simple</span> <span m="3656940">method</span> <span m="3657240">like</span> <span m="3657450">that.</span> <span m="3657870">And</span> <span m="3658290">let</span> <span m="3658470">me</span> <span m="3658560">show</span> <span m="3658800">you</span> <span m="3658950">an</span> <span m="3659030">example</span> <span m="3660590">where</span> <span m="3660890">it</span> <span m="3662830">is</span> <span m="3663620">way</span> <span m="3663950">off.</span> </p>
<p><span m="3674310">All</span> <span m="3674460">right,</span> <span m="3674620">say</span> <span m="3674790">you</span> <span m="3674950">were</span> <span m="3675120">just</span> <span m="3675380">a</span> <span m="3675420">little</span> <span m="3675730">more</span> <span m="3677210">sophisticated</span> <span m="3677980">about</span> <span m="3678310">this.</span> <span m="3679350">And</span> <span m="3679650">when</span> <span m="3679810">you</span> <span m="3680950">did</span> <span m="3681150">your</span> <span m="3681330">observations,</span> <span m="3682010">you</span> <span m="3682090">tried</span> <span m="3682360">to</span> <span m="3682420">figure</span> <span m="3682780">out</span> <span m="3683370">what</span> <span m="3683580">this</span> <span m="3683750">curve</span> <span m="3684200">really</span> <span m="3685010">looks</span> <span m="3685310">like.</span> <span m="3686930">And</span> <span m="3687120">say</span> <span m="3687410">that</span> <span m="3687560">you,</span> <span m="3688310">looking</span> <span m="3688630">at</span> <span m="3688700">the</span> <span m="3688770">data,</span> <span m="3689090">concluded</span> <span m="3689660">that</span> <span m="3691210">the</span> <span m="3691310">probability</span> <span m="3692020">that</span> <span m="3692320">you</span> <span m="3692410">have</span> <span m="3692510">a</span> <span m="3692550">delay</span> <span m="3692970">of</span> <span m="3693070">more</span> <span m="3693330">than</span> <span m="3693470">i</span> <span m="3693650">milliseconds</span> <span m="3695210">is</span> <span m="3695410">1</span> <span m="3695610">and</span> <span m="3695730">i.</span> <span m="3697610">That</span> <span m="3697790">looks</span> <span m="3698050">like</span> <span m="3698280">this</span> <span m="3698460">curve,</span> <span m="3698810">as</span> <span m="3699310">close</span> <span m="3699520">as</span> <span m="3699940">anything.</span> </p>
<p><span m="3701480">So</span> <span m="3701670">from</span> <span m="3701950">your</span> <span m="3702070">data,</span> <span m="3702400">you</span> <span m="3702540">conclude</span> <span m="3702970">this.</span> <span m="3705310">That's</span> <span m="3705560">a</span> <span m="3705730">little</span> <span m="3705770">more</span> <span m="3705930">sophisticated</span> <span m="3706580">conclusion,</span> <span m="3707150">because</span> <span m="3707340">now</span> <span m="3707660">you've</span> <span m="3707850">identified</span> <span m="3709160">what</span> <span m="3709300">you</span> <span m="3709380">believe</span> <span m="3709640">the</span> <span m="3709700">distribution</span> <span m="3710340">is,</span> <span m="3711152">which</span> <span m="3711560">is</span> <span m="3711640">stronger</span> <span m="3712060">than</span> <span m="3712740">expectation.</span> <span m="3715000">How</span> <span m="3715290">would</span> <span m="3715440">you</span> <span m="3715550">go</span> <span m="3715730">about</span> <span m="3716000">figuring</span> <span m="3716360">out</span> <span m="3716480">the</span> <span m="3716560">expected</span> <span m="3717050">value</span> <span m="3719020">if</span> <span m="3719170">you</span> <span m="3719280">had</span> <span m="3719450">that</span> <span m="3719640">information?</span> <span m="3723800">I</span> <span m="3723870">mean,</span> <span m="3723980">you</span> <span m="3724050">could</span> <span m="3724300">average</span> <span m="3724650">the</span> <span m="3725370">100</span> <span m="3725740">sample</span> <span m="3726070">points.</span> <span m="3726380">But</span> <span m="3727730">is</span> <span m="3727870">there</span> <span m="3727980">any</span> <span m="3728120">way,</span> <span m="3728370">if</span> <span m="3729260">you</span> <span m="3729540">assume</span> <span m="3729970">this,</span> <span m="3730240">what</span> <span m="3730370">would</span> <span m="3730480">you</span> <span m="3730550">do</span> <span m="3731530">for the</span> <span m="3731670">expected</span> <span m="3732070">value?</span> </p>
<p><span m="3734680">Yeah,</span> <span m="3735040">did</span> <span m="3735160">I</span> <span m="3735260">erase</span> <span m="3735530">the</span> <span m="3735600">theorem?</span> <span m="3736500">No,</span> <span m="3736760">it's</span> <span m="3736890">over</span> <span m="3737040">there.</span> <span m="3738870">We</span> <span m="3739090">just</span> <span m="3739250">plug</span> <span m="3739480">it</span> <span m="3739550">to</span> <span m="3739740">the</span> <span m="3739820">theorem.</span> <span m="3742190">The</span> <span m="3742270">expected</span> <span m="3742790">delay</span> <span m="3743160">is</span> <span m="3743270">going</span> <span m="3743400">to</span> <span m="3743470">be</span> <span m="3743590">the</span> <span m="3743700">sum</span> <span m="3744080">of</span> <span m="3744230">those</span> <span m="3744460">probabilities</span> <span m="3747590">from</span> <span m="3748930">1</span> <span m="3749160">to</span> <span m="3749280">infinity.</span> <span m="3749800">So</span> <span m="3749970">we</span> <span m="3750080">would</span> <span m="3750210">compute</span> <span m="3751980">from</span> <span m="3752170">the</span> <span m="3752230">theorem</span> <span m="3752760">the</span> <span m="3752830">expected</span> <span m="3753360">delay</span> <span m="3755990">is</span> <span m="3756190">i</span> <span m="3756410">equals</span> <span m="3756790">1</span> <span m="3757190">to</span> <span m="3757390">infinity</span> <span m="3759150">probability.</span> <span m="3763090">Do I</span> <span m="3763190">want</span> <span m="3763250">to</span> <span m="3763310">do</span> <span m="3763430">the</span> <span m="3763540">0</span> <span m="3764040">case?</span> <span m="3765440">Want</span> <span m="3765540">to</span> <span m="3765600">be</span> <span m="3765660">sure</span> <span m="3765820">I</span> <span m="3765850">don't</span> <span m="3765980">get</span> <span m="3766090">caught</span> <span m="3766270">up</span> <span m="3766340">in</span> <span m="3766400">the</span> <span m="3766470">0</span> <span m="3766750">case.</span> <span m="3766970">We'll</span> <span m="3767070">use</span> <span m="3767240">the</span> <span m="3767300">case</span> <span m="3767570">1</span> <span m="3767920">to</span> <span m="3768040">infinity.</span> <span m="3768560">Probability</span> <span m="3769380">D</span> <span m="3769720">greater</span> <span m="3769825">than</span> <span m="3769930">or equal</span> <span m="3770310">to i.</span> <span m="3771070">So</span> <span m="3771230">let me</span> <span m="3771540">put</span> <span m="3771740">greater</span> <span m="3772000">than</span> <span m="3772130">or</span> <span m="3772200">equal</span> <span m="3772390">to</span> <span m="3772570">here.</span> <span m="3776740">That</span> <span m="3777000">equals</span> <span m="3777330">the</span> <span m="3777420">sum</span> <span m="3777705">i</span> <span m="3778410">equals</span> <span m="3778780">1</span> <span m="3779020">to</span> <span m="3779150">infinity</span> <span m="3780170">of</span> <span m="3780320">1</span> <span m="3780520">over</span> <span m="3780780">i.</span> </p>
<p><span m="3782220">What's</span> <span m="3782720">that?</span> <span m="3787220">What's</span> <span m="3787440">the</span> <span m="3787530">sum</span> <span m="3787890">of</span> <span m="3788010">1</span> <span m="3788220">over</span> <span m="3788450">i,</span> <span m="3788960">i</span> <span m="3789160">from</span> <span m="3789360">1</span> <span m="3789530">to</span> <span m="3789640">infinity?</span> <span m="3791630">It's</span> <span m="3791840">infinite.</span> <span m="3793290">Remember</span> <span m="3793480">those</span> <span m="3793680">harmonic</span> <span m="3794160">number?</span> <span m="3795830">i</span> <span m="3796150">going</span> <span m="3796400">from</span> <span m="3796560">1</span> <span m="3796880">to</span> <span m="3797020">n</span> <span m="3797730">gives</span> <span m="3797890">you</span> <span m="3797950">about</span> <span m="3798210">log</span> <span m="3798420">n.</span> <span m="3799120">Remember</span> <span m="3799270">that,</span> <span m="3799870">the</span> <span m="3799960">book</span> <span m="3800170">stacking</span> <span m="3800560">thing?</span> <span m="3802290">i</span> <span m="3802790">going</span> <span m="3802970">from</span> <span m="3803100">1</span> <span m="3803270">to</span> <span m="3803370">infinity,</span> <span m="3805380">that's</span> <span m="3805640">infinity.</span> </p>
<p><span m="3808620">So</span> <span m="3808790">in</span> <span m="3808900">fact,</span> <span m="3809330">your</span> <span m="3809450">expected</span> <span m="3810080">latency</span> <span m="3810760">is</span> <span m="3810890">infinite.</span> <span m="3813070">And</span> <span m="3813240">you</span> <span m="3813330">just</span> <span m="3813520">published</span> <span m="3813870">a</span> <span m="3813920">paper</span> <span m="3814280">saying</span> <span m="3814590">it</span> <span m="3814690">was</span> <span m="3814830">10</span> <span m="3815070">milliseconds,</span> <span m="3815720">maybe</span> <span m="3815970">nine.</span> <span m="3821020">So</span> <span m="3821180">it's</span> <span m="3821270">very</span> <span m="3821560">dangerous</span> <span m="3822150">to</span> <span m="3822260">just</span> <span m="3822460">take</span> <span m="3822630">a</span> <span m="3822670">bunch</span> <span m="3822910">of</span> <span m="3822980">the</span> <span m="3823050">points,</span> <span m="3823460">add</span> <span m="3823560">them</span> <span m="3823810">up,</span> <span m="3823980">and</span> <span m="3824060">average</span> <span m="3824420">them</span> <span m="3825180">and</span> <span m="3825510">say</span> <span m="3825720">that</span> <span m="3825980">is</span> <span m="3826120">the</span> <span m="3826260">answer.</span> <span m="3828060">Especially</span> <span m="3828630">if</span> <span m="3828700">you</span> <span m="3828770">have</span> <span m="3828860">some</span> <span m="3829070">reason</span> <span m="3829320">to</span> <span m="3829380">believe</span> <span m="3829770">the</span> <span m="3829860">distribution</span> <span m="3830500">looks</span> <span m="3830690">like</span> <span m="3830870">this</span> <span m="3831130">and</span> <span m="3831250">it</span> <span m="3831390">really</span> <span m="3831620">is</span> <span m="3831700">something</span> <span m="3831970">like</span> <span m="3832160">that.</span> <span m="3832370">It</span> <span m="3832440">could</span> <span m="3832560">be</span> <span m="3832680">infinite.</span> </p>
<p><span m="3834870">Now</span> <span m="3835010">in</span> <span m="3835080">some</span> <span m="3835440">cases,</span> <span m="3835910">if</span> <span m="3836000">your</span> <span m="3836090">distribution</span> <span m="3836590">is</span> <span m="3836660">very</span> <span m="3836840">well</span> <span m="3837010">behaved,</span> <span m="3837790">averaging</span> <span m="3838290">your</span> <span m="3838340">sample</span> <span m="3838680">points</span> <span m="3838970">is</span> <span m="3839060">the</span> <span m="3839140">perfect</span> <span m="3839490">thing</span> <span m="3839630">to</span> <span m="3839700">do.</span> <span m="3841320">But</span> <span m="3841600">it</span> <span m="3841820">helps</span> <span m="3842100">to</span> <span m="3842200">know</span> <span m="3843210">it's</span> <span m="3843380">not</span> <span m="3843560">necessarily</span> <span m="3844040">the</span> <span m="3844120">case</span> <span m="3844730">that's</span> <span m="3845183">a good way</span> <span m="3845636">to go.</span> <span m="3849000">Any</span> <span m="3849190">ideas</span> <span m="3849490">what</span> <span m="3849620">went</span> <span m="3849800">wrong?</span> <span m="3850530">Yeah.</span> </p>
<p><span m="3851018">AUDIENCE: I have</span> <span m="3851506">a question.</span> <span m="3851994">What would be your</span> <span m="3852482">probability be</span> <span m="3852970">over i</span> <span m="3853410">squared?</span> </p>
<p><span m="3854290">PROFESSOR: Yeah,</span> <span m="3854470">what</span> <span m="3854620">would</span> <span m="3854720">happen</span> <span m="3855050">then?</span> <span m="3855530">If</span> <span m="3855660">in</span> <span m="3855780">fact,</span> <span m="3858490">it</span> <span m="3858540">was</span> <span m="3858690">1</span> <span m="3858920">over</span> <span m="3859160">i</span> <span m="3859280">squared.</span> <span m="3863840">So</span> <span m="3863970">you</span> <span m="3864070">need</span> <span m="3864210">to</span> <span m="3864270">do</span> <span m="3864400">this</span> <span m="3864590">sum.</span> <span m="3866590">Here's</span> <span m="3866760">a</span> <span m="3866820">good</span> <span m="3866990">review</span> <span m="3867300">question</span> <span m="3867710">for</span> <span m="3867810">the</span> <span m="3867880">final.</span> <span m="3868290">What</span> <span m="3868470">method</span> <span m="3868790">do you</span> <span m="3868970">use</span> <span m="3869240">to</span> <span m="3869330">estimate</span> <span m="3869750">that?</span> <span m="3872770">Remember</span> <span m="3873000">the</span> <span m="3873110">how</span> <span m="3873210">do</span> <span m="3873270">we</span> <span m="3873360">do</span> <span m="3873490">that?</span> <span m="3875360">Is</span> <span m="3875500">that</span> <span m="3875670">infinite?</span> <span m="3877540">No.</span> <span m="3878580">He</span> <span m="3878700">used</span> <span m="3879050">the</span> <span m="3879200">integration</span> <span m="3879740">bound.</span> </p>
<p><span m="3881250">And</span> <span m="3881400">you'll</span> <span m="3881520">see</span> <span m="3881850">what</span> <span m="3882040">this</span> <span m="3882260">is</span> <span m="3883580">pretty</span> <span m="3883780">small.</span> <span m="3885330">It'll</span> <span m="3885420">be,</span> <span m="3885590">what,</span> <span m="3885780">1</span> <span m="3885810">and</span> <span m="3885870">a</span> <span m="3885930">1/2,</span> <span m="3886310">2,</span> <span m="3886550">something</span> <span m="3886780">like</span> <span m="3886910">that</span> <span m="3887130">where we</span> <span m="3887260">can do</span> <span m="3887400">the</span> <span m="3887540">integration</span> <span m="3888200">method</span> <span m="3888470">here.</span> <span m="3889660">So</span> <span m="3889900">huge</span> <span m="3890260">difference.</span> <span m="3891890">This</span> <span m="3892050">is</span> <span m="3892170">O</span> <span m="3892330">of</span> <span m="3892410">1.</span> <span m="3892740">This</span> <span m="3892920">is</span> <span m="3893010">bounded.</span> <span m="3894190">Probably</span> <span m="3894430">less</span> <span m="3894630">than</span> <span m="3894740">2,</span> <span m="3895660">if</span> <span m="3895760">we</span> <span m="3895830">did</span> <span m="3895970">the</span> <span m="3896040">integration</span> <span m="3896490">method.</span> <span m="3898760">So</span> <span m="3898870">huge</span> <span m="3899380">difference</span> <span m="3899770">here</span> <span m="3900780">in</span> <span m="3900920">what</span> <span m="3901050">the</span> <span m="3901160">outcome</span> <span m="3901490">is.</span> </p>
<p><span m="3902650">Now,</span> <span m="3903790">how</span> <span m="3904120">can</span> <span m="3904270">it</span> <span m="3904380">be</span> <span m="3906390">that</span> <span m="3906550">I've</span> <span m="3906670">got</span> <span m="3906900">something</span> <span m="3907120">with</span> <span m="3907240">expected</span> <span m="3907950">infinite</span> <span m="3908400">value</span> <span m="3909780">and</span> <span m="3909940">I</span> <span m="3910050">average</span> <span m="3910440">100</span> <span m="3911010">points</span> <span m="3912150">and</span> <span m="3912280">I</span> <span m="3912340">got</span> <span m="3912550">10</span> <span m="3912760">milliseconds?</span> <span m="3914180">Yeah.</span> </p>
<p><span m="3916600">AUDIENCE: The</span> <span m="3916750">infinite</span> <span m="3917120">value</span> <span m="3917456">comes from</span> <span m="3917792">the fact</span> <span m="3918130">that</span> <span m="3919050">there</span> <span m="3919340">is</span> <span m="3919620">a</span> <span m="3920071">decent</span> <span m="3920522">probability</span> <span m="3920973">that</span> <span m="3921424">delay</span> <span m="3921875">is</span> <span m="3921987">going</span> <span m="3922100">to</span> <span m="3922213">be</span> <span m="3922326">huge.</span> <span m="3923228">However,</span> <span m="3923679">if you</span> <span m="3924130">only</span> <span m="3924510">take a</span> <span m="3924840">finite</span> <span m="3925160">number of</span> <span m="3925360">sample</span> <span m="3925833">points,</span> <span m="3926306">then</span> <span m="3927252">chances</span> <span m="3927725">are</span> <span m="3928198">you're not</span> <span m="3928671">going to get</span> <span m="3929144">any</span> <span m="3929617">monstrous</span> <span m="3930090">delays.</span> </p>
<p><span m="3930790">PROFESSOR: Exactly.</span> <span m="3932010">The</span> <span m="3932130">chance</span> <span m="3932520">of</span> <span m="3932620">seeing</span> <span m="3932890">anything</span> <span m="3933220">beyond</span> <span m="3933550">100</span> <span m="3933940">milliseconds</span> <span m="3934670">is</span> <span m="3935470">1</span> <span m="3935690">in</span> <span m="3935760">100.</span> <span m="3937900">So</span> <span m="3938010">I'm</span> <span m="3938120">probably</span> <span m="3938510">not</span> <span m="3938950">going</span> <span m="3939080">to</span> <span m="3939150">see.</span> <span m="3939260">In</span> <span m="3939340">a</span> <span m="3939380">sample</span> <span m="3939760">size of</span> <span m="3940060">100,</span> <span m="3940670">almost</span> <span m="3941140">surely</span> <span m="3941530">I</span> <span m="3941600">won't</span> <span m="3941860">see</span> <span m="3942020">something</span> <span m="3943180">that</span> <span m="3943310">takes</span> <span m="3943550">a</span> <span m="3943610">second,</span> <span m="3944280">1,000</span> <span m="3944680">milliseconds.</span> <span m="3946550">But</span> <span m="3946770">yet</span> <span m="3946990">it's</span> <span m="3947160">those</span> <span m="3947620">rare</span> <span m="3948200">sample</span> <span m="3948600">points</span> <span m="3949470">that</span> <span m="3949600">are</span> <span m="3949640">causing</span> <span m="3950060">that</span> <span m="3950280">sum</span> <span m="3950630">to</span> <span m="3950720">blow</span> <span m="3951000">up.</span> <span m="3952620">If</span> <span m="3952650">I</span> <span m="3952770">sum</span> <span m="3953110">that</span> <span m="3953280">from</span> <span m="3953430">1</span> <span m="3953610">to</span> <span m="3953700">100,</span> <span m="3954110">I</span> <span m="3954170">get</span> <span m="3954360">log</span> <span m="3954630">of</span> <span m="3954720">10.</span> <span m="3956150">Pretty</span> <span m="3956330">small.</span> <span m="3959300">So</span> <span m="3959430">I</span> <span m="3959570">get</span> <span m="3959710">log</span> <span m="3959950">of 100,</span> <span m="3961260">which</span> <span m="3961400">is</span> <span m="3961490">pretty</span> <span m="3961680">small.</span> </p>
<p><span m="3964150">So</span> <span m="3964340">what's</span> <span m="3964520">happening</span> <span m="3964950">when</span> <span m="3965050">you</span> <span m="3965120">do</span> <span m="3965230">your</span> <span m="3965360">finite</span> <span m="3965770">sample</span> <span m="3966210">is</span> <span m="3966290">you're</span> <span m="3966400">missing</span> <span m="3967910">the</span> <span m="3968010">big</span> <span m="3968300">guys</span> <span m="3968760">which</span> <span m="3968950">are</span> <span m="3969000">very</span> <span m="3969300">rare,</span> <span m="3969770">but</span> <span m="3969900">enough</span> <span m="3970170">to</span> <span m="3970260">blow</span> <span m="3970580">up</span> <span m="3970700">your</span> <span m="3970820">expectation.</span> <span m="3972130">Now,</span> <span m="3972290">you</span> <span m="3972400">can</span> <span m="3972580">draw</span> <span m="3972820">two</span> <span m="3973010">conclusions</span> <span m="3973600">from</span> <span m="3973720">that.</span> <span m="3975020">One</span> <span m="3975320">of</span> <span m="3975380">them</span> <span m="3975480">is</span> <span m="3975560">just</span> <span m="3975790">we</span> <span m="3975920">did.</span> <span m="3976250">The</span> <span m="3976340">other</span> <span m="3976560">is,</span> <span m="3976720">well,</span> <span m="3976950">expectation's</span> <span m="3977640">the</span> <span m="3977700">wrong</span> <span m="3977980">measure.</span> <span m="3978830">And</span> <span m="3979020">really</span> <span m="3979230">what</span> <span m="3979330">we</span> <span m="3979390">should</span> <span m="3979510">be</span> <span m="3979590">doing</span> <span m="3979950">is</span> <span m="3980060">looking</span> <span m="3980320">at</span> <span m="3980400">only</span> <span m="3981380">1,000</span> <span m="3981840">sample</span> <span m="3982200">points</span> <span m="3983130">or</span> <span m="3983270">something</span> <span m="3983600">like</span> <span m="3983770">that.</span> </p>
<p><span m="3984020">But</span> <span m="3984140">in</span> <span m="3984220">practice</span> <span m="3984650">of</span> <span m="3984720">using</span> <span m="3984970">the</span> <span m="3985030">thing</span> <span m="3985190">over</span> <span m="3985370">and</span> <span m="3985460">over</span> <span m="3985640">again,</span> <span m="3986230">eventually you're</span> <span m="3986610">going</span> <span m="3986770">to</span> <span m="3986840">get</span> <span m="3986960">hit</span> <span m="3987100">with</span> <span m="3987220">a</span> <span m="3987260">whopper.</span> <span m="3988480">Sometimes</span> <span m="3988880">you'll</span> <span m="3988980">see</span> <span m="3989140">people</span> <span m="3989410">take</span> <span m="3989820">data</span> <span m="3990100">points</span> <span m="3990420">out</span> <span m="3990630">when</span> <span m="3990750">they're</span> <span m="3990860">really</span> <span m="3991100">the</span> <span m="3991180">big</span> <span m="3991380">ones.</span> <span m="3992730">They say,</span> <span m="3992900">oh,</span> <span m="3993120">well,</span> <span m="3993220">that</span> <span m="3993350">was</span> <span m="3993450">an</span> <span m="3993520">anomaly.</span> <span m="3994520">I</span> <span m="3994630">take</span> <span m="3994860">that</span> <span m="3995050">out</span> <span m="3995250">and</span> <span m="3995340">then</span> <span m="3995440">we</span> <span m="3995530">compute</span> <span m="3995860">the</span> <span m="3995980">average.</span> <span m="3997330">Yeah?</span> </p>
<p><span m="3997792">AUDIENCE: How</span> <span m="3998254">much would you</span> <span m="3998716">pay to</span> <span m="3999178">play a game</span> <span m="3999640">where</span> <span m="3999670">the pair would</span> <span m="4000162">the be the</span> <span m="4000654">latency</span> <span m="4001146">of the</span> <span m="4001638">packet?</span> </p>
<p><span m="4003120">PROFESSOR: What's</span> <span m="4003580">that?</span> </p>
<p><span m="4004073">AUDIENCE: How much</span> <span m="4004566">do you  pay to</span> <span m="4005060">play a game</span> <span m="4005491">where the</span> <span m="4005922">pair would</span> <span m="4006353">be the</span> <span m="4006784">latency</span> <span m="4007215">of the packet?</span> </p>
<p><span m="4008080">PROFESSOR: All</span> <span m="4008320">right.</span> <span m="4011720">That's</span> <span m="4012020">a</span> <span m="4012070">big</span> <span m="4012260">number.</span> <span m="4013810">If</span> <span m="4013940">that</span> <span m="4014090">was</span> <span m="4014230">my</span> <span m="4014760">losses</span> <span m="4015300">here,</span> <span m="4016280">that's</span> <span m="4016670">tough.</span> <span m="4017170">I'd</span> <span m="4018300">bet</span> <span m="4018590">anything</span> <span m="4019020">up</span> <span m="4019140">against</span> <span m="4019440">that.</span> <span m="4020500">To</span> <span m="4020600">get</span> <span m="4020740">a</span> <span m="4020780">payoff</span> <span m="4021160">of</span> <span m="4021320">infinity?</span> <span m="4022730">You'd</span> <span m="4022880">pay</span> <span m="4023070">$1,000</span> <span m="4023930">to</span> <span m="4024000">play</span> <span m="4024390">that</span> <span m="4024570">game.</span> <span m="4025260">Now,</span> <span m="4025380">you'd</span> <span m="4025520">want</span> <span m="4025700">to</span> <span m="4025760">play</span> <span m="4025990">it</span> <span m="4026080">for</span> <span m="4026190">a</span> <span m="4026260">long</span> <span m="4026840">time</span> <span m="4028040">to</span> <span m="4028160">get</span> <span m="4028370">that</span> <span m="4028550">big</span> <span m="4028750">payoff,</span> <span m="4029240">right?</span> <span m="4031050">But</span> <span m="4031200">eventually</span> <span m="4032330">that's</span> <span m="4032600">what</span> <span m="4032730">it's</span> <span m="4032820">going</span> <span m="4032950">to</span> <span m="4033020">be.</span> <span m="4039260">Any</span> <span m="4039430">questions?</span> <span m="4039860">People</span> <span m="4040100">understand</span> <span m="4040640">the</span> <span m="4041360">issue</span> <span m="4041680">here</span> <span m="4042790">and</span> <span m="4042920">what</span> <span m="4043040">to</span> <span m="4043140">worry</span> <span m="4043410">about</span> <span m="4043810">when</span> <span m="4044770">you</span> <span m="4044850">all</span> <span m="4044970">do</span> <span m="4045100">that</span> <span m="4045280">someday?</span> <span m="4045650">Yeah.</span> </p>
<p><span m="4046650">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="4053350">PROFESSOR: Yeah,</span> <span m="4054700">now</span> <span m="4054810">here</span> <span m="4055030">you</span> <span m="4055110">don't</span> <span m="4055280">know</span> <span m="4055460">for</span> <span m="4055620">a</span> <span m="4055670">fact</span> <span m="4056120">this</span> <span m="4056270">is</span> <span m="4056400">it.</span> <span m="4056600">But</span> <span m="4056750">you</span> <span m="4056840">could</span> <span m="4057000">model</span> <span m="4057430">and</span> <span m="4057530">you</span> <span m="4057610">start</span> <span m="4057870">seeing</span> <span m="4058430">these.</span> <span m="4058640">You</span> <span m="4058790">fill</span> <span m="4059090">in</span> <span m="4059970">the</span> <span m="4060070">points</span> <span m="4060420">and you</span> <span m="4060570">say,</span> <span m="4060750">it</span> <span m="4060940">looks</span> <span m="4061320">like</span> <span m="4061550">this,</span> <span m="4062160">let's</span> <span m="4062340">assume</span> <span m="4062610">that's</span> <span m="4062820">what</span> <span m="4062930">it</span> <span m="4063010">was,</span> <span m="4063900">then</span> <span m="4064100">here's</span> <span m="4064360">the</span> <span m="4064430">result.</span> <span m="4066010">If</span> <span m="4066110">it</span> <span m="4066200">looks</span> <span m="4066430">like</span> <span m="4066550">it's 1 over i</span> <span m="4066620">squared,</span> <span m="4067090">you can say,</span> <span m="4067470">let's</span> <span m="4067650">assume</span> <span m="4068040">that's</span> <span m="4068240">what</span> <span m="4068360">it</span> <span m="4068440">is</span> <span m="4069350">and</span> <span m="4069460">then</span> <span m="4069550">you</span> <span m="4069610">get</span> <span m="4069700">a</span> <span m="4069730">different</span> <span m="4070050">result.</span> <span m="4070510">But</span> <span m="4070610">you</span> <span m="4071590">take</span> <span m="4072080">the</span> <span m="4072280">various</span> <span m="4072620">cases</span> <span m="4073050">and</span> <span m="4073150">consider</span> <span m="4073520">them</span> <span m="4074650">to</span> <span m="4074770">do</span> <span m="4074970">it.</span> </p>
<p><span m="4075210">Or</span> <span m="4075660">you</span> <span m="4075760">could</span> <span m="4075900">say,</span> <span m="4077090">I</span> <span m="4077250">take</span> <span m="4077480">the</span> <span m="4077570">expectation</span> <span m="4078310">assuming</span> <span m="4078910">I</span> <span m="4078960">never</span> <span m="4079200">get</span> <span m="4079370">a</span> <span m="4079400">point</span> <span m="4080050">bigger</span> <span m="4080340">than</span> <span m="4080860">100.</span> <span m="4082270">And</span> <span m="4082510">I</span> <span m="4082580">limit</span> <span m="4082960">it</span> <span m="4083050">that</span> <span m="4083290">way.</span> <span m="4083540">And</span> <span m="4083640">then</span> <span m="4083650">you're</span> <span m="4083770">safe</span> <span m="4084240">at</span> <span m="4084370">that</span> <span m="4084530">point</span> <span m="4085300">and</span> <span m="4085330">you</span> <span m="4085400">can</span> <span m="4085540">get</span> <span m="4085660">away</span> <span m="4085850">with</span> <span m="4085990">it.</span> <span m="4086450">But</span> <span m="4086640">to</span> <span m="4086710">blindly</span> <span m="4087210">go</span> <span m="4087350">out</span> <span m="4087550">there</span> <span m="4087680">and</span> <span m="4087770">say</span> <span m="4087940">here</span> <span m="4088140">it</span> <span m="4088220">is,</span> <span m="4088980">not</span> <span m="4089260">so</span> <span m="4089660">reliable.</span> </p>
<p><span m="4096359">The</span> <span m="4096450">expected</span> <span m="4096950">value</span> <span m="4097229">does</span> <span m="4097729">have</span> <span m="4097880">a</span> <span m="4097920">lot</span> <span m="4098180">of</span> <span m="4098260">useful</span> <span m="4098569">properties.</span> <span m="4100410">And</span> <span m="4101340">the</span> <span m="4101450">most</span> <span m="4101979">important</span> <span m="4103560">is</span> <span m="4103750">called</span> <span m="4104000">linearity</span> <span m="4104529">of</span> <span m="4104630">expectation.</span> <span m="4105510">And</span> <span m="4105640">we'll</span> <span m="4105729">spend</span> <span m="4105950">the</span> <span m="4106029">rest</span> <span m="4106290">of</span> <span m="4106359">today</span> <span m="4106840">and</span> <span m="4106990">some</span> <span m="4107120">of</span> <span m="4107200">next</span> <span m="4107657">time talking</span> <span m="4108114">about it.</span> <span m="4112229">And</span> <span m="4113149">quite</span> <span m="4113399">possibly</span> <span m="4113930">it's</span> <span m="4114090">one</span> <span m="4114220">of</span> <span m="4114279">the</span> <span m="4114359">reasons</span> <span m="4114660">people</span> <span m="4114910">use</span> <span m="4115250">it</span> <span m="4115370">so</span> <span m="4115520">much</span> <span m="4115779">instead</span> <span m="4116020">of</span> <span m="4116120">other</span> <span m="4116250">things</span> <span m="4116529">you</span> <span m="4116600">might</span> <span m="4117340">think</span> <span m="4117590">about.</span> </p>
<p><span m="4133020">The</span> <span m="4133140">theorem,</span> <span m="4133600">and</span> <span m="4133710">this</span> <span m="4134010">may</span> <span m="4134300">be</span> <span m="4134430">the</span> <span m="4134510">most</span> <span m="4134790">important</span> <span m="4135170">theorem</span> <span m="4135319">on</span> <span m="4135490">probability.</span> <span m="4138810">For</span> <span m="4139000">any</span> <span m="4139450">random</span> <span m="4139859">variables,</span> <span m="4141460">R1</span> <span m="4142859">and</span> <span m="4143010">R2,</span> <span m="4146020">on</span> <span m="4146370">a</span> <span m="4146450">probability</span> <span m="4146970">space</span> <span m="4147350">S,</span> <span m="4153420">the</span> <span m="4153550">expected</span> <span m="4154140">value</span> <span m="4156160">of</span> <span m="4156340">R1</span> <span m="4156870">plus</span> <span m="4157240">R2</span> <span m="4158819">equals</span> <span m="4159310">the</span> <span m="4159399">expected</span> <span m="4159910">value</span> <span m="4160270">of</span> <span m="4160380">R1</span> <span m="4161890">plus</span> <span m="4162319">the</span> <span m="4162420">expected</span> <span m="4162910">value</span> <span m="4163859">of</span> <span m="4164029">R2.</span> <span m="4166290">Very</span> <span m="4166790">simple.</span> <span m="4167649">It's</span> <span m="4167800">another</span> <span m="4167899">way</span> <span m="4168010">of</span> <span m="4168080">saying</span> <span m="4168370">expectation</span> <span m="4169130">is</span> <span m="4169220">a</span> <span m="4169290">linear</span> <span m="4170080">function.</span> </p>
<p><span m="4172240">The</span> <span m="4172319">proof.</span> <span m="4177109">No,</span> <span m="4177340">skip</span> <span m="4177740">the</span> <span m="4177830">proof</span> <span m="4178160">here.</span> <span m="4178330">It's</span> <span m="4178460">not</span> <span m="4178670">hard</span> <span m="4179107">and it's in</span> <span m="4179544">the text.</span> <span m="4183760">Follows</span> <span m="4184410">pretty</span> <span m="4184649">simply</span> <span m="4185054">from the</span> <span m="4185459">definition.</span> <span m="4193250">There's</span> <span m="4193479">a</span> <span m="4193529">generalization</span> <span m="4194510">for</span> <span m="4194650">more</span> <span m="4195090">than</span> <span m="4196050">two</span> <span m="4196290">random</span> <span m="4196610">variables.</span> <span m="4199810">So</span> <span m="4200520">corollary.</span> <span m="4203990">For</span> <span m="4204200">all</span> <span m="4204460">k</span> <span m="4204950">in</span> <span m="4205030">the</span> <span m="4205100">natural</span> <span m="4205470">numbers</span> <span m="4207950">and</span> <span m="4208480">k</span> <span m="4208770">random</span> <span m="4209190">variables</span> <span m="4211130">R1,</span> <span m="4212390">R2,</span> <span m="4214180">up</span> <span m="4214340">to</span> <span m="4214430">Rk</span> <span m="4217070">on</span> <span m="4217340">the</span> <span m="4217410">sample</span> <span m="4218160">probability</span> <span m="4218560">space</span> <span m="4218920">S.</span> <span m="4220680">The</span> <span m="4220820">expected</span> <span m="4221390">value</span> <span m="4222000">of</span> <span m="4222500">R1</span> <span m="4223100">plus</span> <span m="4223470">R2</span> <span m="4225530">plus</span> <span m="4225770">Rk</span> <span m="4227480">is</span> <span m="4227830">just</span> <span m="4228090">the</span> <span m="4228160">sum</span> <span m="4228390">of</span> <span m="4228490">the</span> <span m="4228570">expected</span> <span m="4229010">values.</span> <span m="4236220">And</span> <span m="4236390">the</span> <span m="4236450">proof</span> <span m="4236670">of</span> <span m="4236750">that</span> <span m="4236930">is</span> <span m="4237050">by</span> <span m="4237250">induction</span> <span m="4238430">using</span> <span m="4238770">that</span> <span m="4238970">result.</span> </p>
<p><span m="4240240">And</span> <span m="4240390">the</span> <span m="4240480">really</span> <span m="4240760">important</span> <span m="4241210">thing</span> <span m="4241340">about</span> <span m="4241630">this</span> <span m="4241920">is</span> <span m="4242160">that</span> <span m="4242480">neither</span> <span m="4242900">of</span> <span m="4242970">these</span> <span m="4243190">results</span> <span m="4243690">needs</span> <span m="4243940">independence.</span> <span m="4247370">It</span> <span m="4247490">is</span> <span m="4247610">true</span> <span m="4249660">whether</span> <span m="4250130">or</span> <span m="4250200">not</span> <span m="4250670">the</span> <span m="4250820">Ri</span> <span m="4252160">are</span> <span m="4252300">independent.</span> <span m="4258850">Pretty</span> <span m="4259050">much</span> <span m="4259310">everything</span> <span m="4259680">we</span> <span m="4259780">do</span> <span m="4259910">in</span> <span m="4260000">probability</span> <span m="4261190">to</span> <span m="4261300">manipulate</span> <span m="4261820">random</span> <span m="4262070">variables</span> <span m="4262580">needs</span> <span m="4262880">them</span> <span m="4263040">to</span> <span m="4263110">be</span> <span m="4263290">independent.</span> <span m="4264530">You</span> <span m="4264650">don't</span> <span m="4265120">need</span> <span m="4265340">that</span> <span m="4265540">here.</span> <span m="4266050">And</span> <span m="4266180">that'll</span> <span m="4266360">make</span> <span m="4266540">it</span> <span m="4266620">very</span> <span m="4266840">powerful.</span> </p>
<p><span m="4268650">So</span> <span m="4268770">let's</span> <span m="4268890">do</span> <span m="4269000">an</span> <span m="4269070">example.</span> <span m="4288370">Say</span> <span m="4288690">I</span> <span m="4288810">roll</span> <span m="4291230">two</span> <span m="4291760">fair</span> <span m="4292020">dice.</span> <span m="4295780">Six</span> <span m="4296080">sided</span> <span m="4296370">dice.</span> <span m="4302320">Not</span> <span m="4302640">necessarily</span> <span m="4303250">independent.</span> <span m="4305360">R1</span> <span m="4305950">is</span> <span m="4306090">the</span> <span m="4306240">outcome</span> <span m="4307710">on</span> <span m="4307850">the</span> <span m="4307910">first</span> <span m="4308230">die.</span> <span m="4313990">And</span> <span m="4314450">R2</span> <span m="4316000">will</span> <span m="4316140">be</span> <span m="4316350">the</span> <span m="4316530">outcome</span> <span m="4317340">on</span> <span m="4317590">the</span> <span m="4317660">second</span> <span m="4318060">one.</span> <span m="4320740">And</span> <span m="4320910">I'm</span> <span m="4321020">interested</span> <span m="4321440">in</span> <span m="4321550">the sum</span> <span m="4322050">of</span> <span m="4322120">the</span> <span m="4322190">dice.</span> <span m="4324680">So</span> <span m="4324830">with that</span> <span m="4325070">R it'd be</span> <span m="4325395">R1</span> <span m="4326450">plus</span> <span m="4326770">R2.</span> <span m="4328470">And</span> <span m="4328700">I</span> <span m="4328740">want</span> <span m="4328970">to</span> <span m="4329030">know</span> <span m="4329160">the</span> <span m="4329260">expected</span> <span m="4329810">value</span> <span m="4331010">of</span> <span m="4331220">the</span> <span m="4331290">sum</span> <span m="4331570">of</span> <span m="4331640">the</span> <span m="4331700">dice</span> <span m="4332100">when</span> <span m="4332220">I</span> <span m="4332290">roll</span> <span m="4332520">them.</span> </p>
<p><span m="4336280">Now,</span> <span m="4337150">if</span> <span m="4337230">I</span> <span m="4337310">didn't</span> <span m="4337600">use</span> <span m="4338000">that</span> <span m="4338380">theorem</span> <span m="4340260">I'd</span> <span m="4340710">compute</span> <span m="4341170">the</span> <span m="4341240">tree</span> <span m="4341760">in</span> <span m="4341870">the</span> <span m="4341930">sample</span> <span m="4342340">space.</span> <span m="4342740">I'd</span> <span m="4342800">get</span> <span m="4342950">36</span> <span m="4343760">possible</span> <span m="4344260">outcomes,</span> <span m="4345330">take</span> <span m="4345590">the</span> <span m="4345670">probability</span> <span m="4346240">of</span> <span m="4346340">each.</span> <span m="4347360">It'd</span> <span m="4347490">take</span> <span m="4347740">you a</span> <span m="4347780">little</span> <span m="4347970">while</span> <span m="4348150">to</span> <span m="4348210">do</span> <span m="4348460">it,</span> <span m="4349210">but</span> <span m="4349380">using</span> <span m="4349740">linearity</span> <span m="4349820">of</span> <span m="4350190">expectation,</span> <span m="4350900">it's</span> <span m="4351050">easy.</span> <span m="4353320">It's</span> <span m="4353500">expected</span> <span m="4353940">value</span> <span m="4354170">of</span> <span m="4354250">R1</span> <span m="4354800">plus</span> <span m="4355050">the</span> <span m="4355110">expected</span> <span m="4355550">value</span> <span m="4355790">of</span> <span m="4355880">R2.</span> <span m="4358070">Each</span> <span m="4358370">of</span> <span m="4358480">these</span> <span m="4358850">we</span> <span m="4358970">already</span> <span m="4359200">figured</span> <span m="4359510">out</span> <span m="4359700">is</span> <span m="4359820">3</span> <span m="4360572">and</span> <span m="4360950">1/2.</span> <span m="4363230">And</span> <span m="4363440">so</span> <span m="4363700">the</span> <span m="4363990">answer</span> <span m="4364450">is</span> <span m="4364570">7.</span> <span m="4366330">So</span> <span m="4366480">if</span> <span m="4366550">you</span> <span m="4366660">roll</span> <span m="4366950">a</span> <span m="4366990">pair</span> <span m="4367250">of</span> <span m="4367330">dice,</span> <span m="4368700">whether</span> <span m="4368960">or</span> <span m="4368990">not</span> <span m="4369170">they</span> <span m="4369230">are</span> <span m="4369270">independent,</span> <span m="4370050">the</span> <span m="4370160">expected</span> <span m="4370760">sum</span> <span m="4372120">is</span> <span m="4372280">seven.</span> <span m="4375370">Any</span> <span m="4375510">questions</span> <span m="4375970">about</span> <span m="4376890">linearity</span> <span m="4377020">of</span> <span m="4377350">expectation?</span> <span m="4379270">Yeah.</span> </p>
<p><span m="4380596">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="4388750">PROFESSOR: This</span> <span m="4388950">one?</span> <span m="4390550">[INAUDIBLE].</span> </p>
<p><span m="4391816">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="4394950">PROFESSOR: No,</span> <span m="4395260">I</span> <span m="4395320">mean</span> <span m="4395560">pluses</span> <span m="4396010">here.</span> <span m="4396530">Here?</span> <span m="4398230">I'm</span> <span m="4398480">computing</span> <span m="4398950">the</span> <span m="4399030">sum</span> <span m="4399530">of</span> <span m="4399620">the</span> <span m="4399720">random</span> <span m="4400030">variables.</span> <span m="4403180">So</span> <span m="4403720">this</span> <span m="4403910">could</span> <span m="4404060">be</span> <span m="4404280">a</span> <span m="4404400">1,</span> <span m="4404840">that</span> <span m="4405000">could</span> <span m="4405120">be</span> <span m="4405240">a</span> <span m="4405300">10,</span> <span m="4405670">this</span> <span m="4405810">could</span> <span m="4405920">be</span> <span m="4406030">a</span> <span m="4406080">3.</span> <span m="4407180">So</span> <span m="4407460">I'm</span> <span m="4407840">compute</span> <span m="4408160">the</span> <span m="4408230">expected</span> <span m="4408830">value</span> <span m="4409110">of</span> <span m="4409180">the</span> <span m="4409250">sum,</span> <span m="4409610">just</span> <span m="4409810">like</span> <span m="4409910">when</span> <span m="4410010">I</span> <span m="4410060">rolled</span> <span m="4410320">two,</span> <span m="4410490">dice</span> <span m="4411610">I'm</span> <span m="4411770">taking</span> <span m="4412040">the</span> <span m="4412130">expected</span> <span m="4412780">value</span> <span m="4413160">of</span> <span m="4413270">the</span> <span m="4413330">sum</span> <span m="4413740">of</span> <span m="4413810">the</span> <span m="4413880">dice.</span> <span m="4416370">Any</span> <span m="4416550">other</span> <span m="4416700">questions?</span> </p>
<p><span m="4423850">Now</span> <span m="4423980">we're</span> <span m="4424040">going</span> <span m="4424200">to</span> <span m="4424270">do</span> <span m="4424560">a</span> <span m="4426040">little</span> <span m="4426390">trickier</span> <span m="4426860">problem</span> <span m="4427480">that</span> <span m="4427610">uses</span> <span m="4428640">linearity of</span> <span m="4429101">expectation.</span> <span m="4433250">Yeah?</span> </p>
<p><span m="4434970">AUDIENCE: [INAUDIBLE]</span> <span m="4435980">sets,</span> <span m="4436960">like</span> <span m="4437450">when</span> <span m="4437940">we're looking</span> <span m="4438430">at the</span> <span m="4440390">failure.</span> <span m="4441845">Couldn't</span> <span m="4442320">we add</span> <span m="4443292">the</span> <span m="4443535">two</span> <span m="4443778">cases,</span> <span m="4444750">like</span> <span m="4445236">R1</span> <span m="4445722">being</span> <span m="4446208">a girl and</span> <span m="4446694">R2</span> <span m="4447180">being</span> <span m="4447666">a boy?</span> </p>
<p><span m="4448650">PROFESSOR: Well,</span> <span m="4448820">what</span> <span m="4449020">would</span> <span m="4449140">it</span> <span m="4449240">mean</span> <span m="4450460">to</span> <span m="4450951">sum--</span> <span m="4454880">so</span> <span m="4455010">what</span> <span m="4455200">is</span> <span m="4455330">the</span> <span m="4455440">random</span> <span m="4455760">variable?</span> <span m="4456090">R1</span> <span m="4456620">is</span> <span m="4456820">the</span> <span m="4456930">case</span> <span m="4457350">when</span> <span m="4457440">you</span> <span m="4457500">get</span> <span m="4457610">a</span> <span m="4457650">boy.</span> <span m="4457820">So</span> <span m="4457930">it's</span> <span m="4458020">an</span> <span m="4458100">indicator</span> <span m="4458620">for</span> <span m="4458720">getting</span> <span m="4458930">a</span> <span m="4458990">boy.</span> <span m="4459940">R2</span> <span m="4460340">is</span> <span m="4460470">the</span> <span m="4460570">indicator</span> <span m="4460960">for</span> <span m="4461070">getting</span> <span m="4461280">a</span> <span m="4461350">girl.</span> <span m="4462790">R1</span> <span m="4463190">plus</span> <span m="4463510">R2</span> <span m="4463960">is</span> <span m="4464090">by</span> <span m="4464230">definition</span> <span m="4464750">1</span> <span m="4465150">then,</span> <span m="4465310">because</span> <span m="4465500">you</span> <span m="4465580">got</span> <span m="4465690">a</span> <span m="4465720">boy</span> <span m="4466010">or</span> <span m="4466230">a</span> <span m="4466280">girl.</span> <span m="4466760">One</span> <span m="4466950">of</span> <span m="4467020">them</span> <span m="4467140">had</span> <span m="4467270">to</span> <span m="4467360">happen.</span> <span m="4468330">And</span> <span m="4468450">the</span> <span m="4468510">expected</span> <span m="4468920">value</span> <span m="4469240">would</span> <span m="4469370">be</span> <span m="4469580">one,</span> <span m="4469920">But</span> <span m="4470030">that's</span> <span m="4470240">a</span> <span m="4470290">different</span> <span m="4470650">kind</span> <span m="4470810">of</span> <span m="4470880">game.</span> <span m="4472690">But</span> <span m="4472930">we</span> <span m="4473100">are</span> <span m="4473270">going</span> <span m="4473390">to</span> <span m="4473450">start</span> <span m="4474360">using</span> <span m="4474910">this</span> <span m="4475120">in</span> <span m="4475240">sophisticated</span> <span m="4475880">ways</span> <span m="4476440">to</span> <span m="4476540">make</span> <span m="4476710">calculations</span> <span m="4477340">be</span> <span m="4477450">easier</span> <span m="4478250">for</span> <span m="4478370">things</span> <span m="4478660">like</span> <span m="4478880">that.</span> <span m="4484000">so</span> </p>
<p><span m="4484180">This</span> <span m="4484380">problem</span> <span m="4484940">we</span> <span m="4485030">call</span> <span m="4485390">the</span> <span m="4485480">hat</span> <span m="4485740">check</span> <span m="4485970">problem.</span> <span m="4491560">And</span> <span m="4491700">the</span> <span m="4491800">idea</span> <span m="4492870">here</span> <span m="4493210">behind</span> <span m="4493680">this</span> <span m="4493910">is</span> <span m="4494010">say</span> <span m="4494250">that</span> <span m="4494340">you</span> <span m="4494460">have</span> <span m="4494880">n</span> <span m="4495160">men</span> <span m="4495760">at</span> <span m="4495910">a</span> <span m="4495960">restaurant</span> <span m="4497530">having</span> <span m="4497820">dinner.</span> <span m="4499350">And</span> <span m="4500060">when</span> <span m="4500210">they</span> <span m="4500290">come</span> <span m="4500500">into</span> <span m="4500620">the</span> <span m="4500690">restaurant,</span> <span m="4501110">they</span> <span m="4501220">check</span> <span m="4501530">their</span> <span m="4501670">hats</span> <span m="4502240">in</span> <span m="4502380">the</span> <span m="4502460">coat</span> <span m="4502750">room.</span> </p>
<p><span m="4504360">Then</span> <span m="4505350">something</span> <span m="4505690">goes</span> <span m="4505900">wrong</span> <span m="4506120">in</span> <span m="4506190">the</span> <span m="4506250">coat</span> <span m="4506560">room</span> <span m="4506790">and</span> <span m="4506880">the</span> <span m="4506960">hats</span> <span m="4507320">are</span> <span m="4507430">all</span> <span m="4507860">scrambled</span> <span m="4508540">up</span> <span m="4508730">randomly.</span> <span m="4510200">Let's</span> <span m="4510390">say</span> <span m="4510620">a</span> <span m="4510910">random</span> <span m="4511260">permutation</span> <span m="4511970">of</span> <span m="4512070">the</span> <span m="4512150">hats.</span> <span m="4513010">So</span> <span m="4513180">the</span> <span m="4513250">men</span> <span m="4513490">come</span> <span m="4513710">back</span> <span m="4513930">to</span> <span m="4514010">get</span> <span m="4514240">their</span> <span m="4514380">hats</span> <span m="4515480">and</span> <span m="4515760">they</span> <span m="4515840">get</span> <span m="4515980">a</span> <span m="4516050">random</span> <span m="4516540">hat</span> <span m="4517410">coming</span> <span m="4517700">back.</span> <span m="4519370">So</span> <span m="4519540">each</span> <span m="4519840">man</span> <span m="4523160">gets</span> <span m="4523410">a</span> <span m="4523500">random</span> <span m="4523830">hat</span> <span m="4524050">back</span> <span m="4529160">after</span> <span m="4529520">dinner.</span> </p>
<p><span m="4531220">And</span> <span m="4531470">the</span> <span m="4531530">question</span> <span m="4532010">is,</span> <span m="4532940">what</span> <span m="4533070">is</span> <span m="4533220">the</span> <span m="4533350">expected</span> <span m="4534140">number</span> <span m="4534440">of</span> <span m="4534550">men</span> <span m="4535520">to</span> <span m="4535640">get</span> <span m="4535860">the</span> <span m="4535970">right</span> <span m="4536410">hat</span> <span m="4536890">back?</span> <span m="4539980">So</span> <span m="4540120">we</span> <span m="4540200">let</span> <span m="4540370">R</span> <span m="4540720">be</span> <span m="4540950">the</span> <span m="4541220">random</span> <span m="4541510">variable</span> <span m="4541840">that</span> <span m="4541980">says</span> <span m="4542350">the</span> <span m="4542420">number</span> <span m="4542790">of</span> <span m="4542890">men</span> <span m="4545180">to</span> <span m="4545340">get</span> <span m="4545590">the</span> <span m="4545690">right</span> <span m="4545890">hat,</span> <span m="4548440">their</span> <span m="4548690">happen.</span> <span m="4552000">And</span> <span m="4552410">we</span> <span m="4552500">want</span> <span m="4552760">to</span> <span m="4552820">know</span> <span m="4553120">the</span> <span m="4553240">expected</span> <span m="4553830">value</span> <span m="4554180">of</span> <span m="4554300">R.</span> <span m="4557140">Now,</span> <span m="4557340">from</span> <span m="4557590">the</span> <span m="4557660">definition,</span> <span m="4558380">that's</span> <span m="4558620">just</span> <span m="4558850">the</span> <span m="4558920">sum</span> <span m="4559840">of</span> <span m="4560800">all</span> <span m="4561000">possibilities.</span> <span m="4561570">K</span> <span m="4562020">from</span> <span m="4562250">1</span> <span m="4562560">to</span> <span m="4562660">n.</span> <span m="4564556">K</span> <span m="4565030">times</span> <span m="4565430">the</span> <span m="4565510">probability</span> <span m="4566990">R</span> <span m="4567290">equals</span> <span m="4567650">k.</span> <span m="4569100">So</span> <span m="4569330">using</span> <span m="4569690">one</span> <span m="4569900">of</span> <span m="4569960">the</span> <span m="4570020">definitions,</span> <span m="4572170">we</span> <span m="4572330">could</span> <span m="4572460">compute</span> <span m="4572860">the</span> <span m="4572930">expected</span> <span m="4573370">value</span> <span m="4573710">this</span> <span m="4573970">way.</span> <span m="4574310">In</span> <span m="4574400">fact,</span> <span m="4574770">if</span> <span m="4574870">you</span> <span m="4574970">were</span> <span m="4575070">to</span> <span m="4575650">be</span> <span m="4575780">assigned</span> <span m="4576190">this</span> <span m="4576350">on</span> <span m="4576460">a</span> <span m="4576520">test</span> <span m="4577820">or</span> <span m="4578110">on</span> <span m="4578270">homework,</span> <span m="4578780">that's</span> <span m="4579050">probably</span> <span m="4579450">how</span> <span m="4579600">you'd</span> <span m="4579770">start</span> <span m="4580200">something</span> <span m="4580500">like</span> <span m="4580720">that.</span> </p>
<p><span m="4583560">And</span> <span m="4583790">then,</span> <span m="4584980">well,</span> <span m="4585220">the</span> <span m="4585300">next</span> <span m="4585560">step</span> <span m="4585770">you'd</span> <span m="4585890">take</span> <span m="4586190">would</span> <span m="4586300">be to</span> <span m="4586410">figure</span> <span m="4586660">out</span> <span m="4586800">what's</span> <span m="4587020">the</span> <span m="4587100">probability</span> <span m="4587720">that</span> <span m="4587840">exactly</span> <span m="4588410">K</span> <span m="4588760">men</span> <span m="4590290">get</span> <span m="4590480">the</span> <span m="4590570">right</span> <span m="4590830">hat</span> <span m="4591160">back.</span> <span m="4592980">In</span> <span m="4593040">fact,</span> <span m="4593520">we</span> <span m="4593640">actually</span> <span m="4594020">asked</span> <span m="4594360">this</span> <span m="4594550">once</span> <span m="4594890">before</span> <span m="4595160">we</span> <span m="4595280">started</span> <span m="4595560">doing</span> <span m="4595760">it</span> <span m="4595840">in</span> <span m="4595920">class.</span> <span m="4597430">And</span> <span m="4597520">it</span> <span m="4597580">was</span> <span m="4597740">really</span> <span m="4598470">hard</span> <span m="4599030">if</span> <span m="4599140">you</span> <span m="4599250">went</span> <span m="4599440">down</span> <span m="4599630">this</span> <span m="4599790">path.</span> <span m="4600170">Because</span> <span m="4600500">if</span> <span m="4600600">you</span> <span m="4600690">spent</span> <span m="4601420">all</span> <span m="4601800">night</span> <span m="4602300">with</span> <span m="4602480">your</span> <span m="4602600">buddies,</span> <span m="4604230">you</span> <span m="4604450">would</span> <span m="4605370">maybe</span> <span m="4605790">get</span> <span m="4605960">to</span> <span m="4606060">the</span> <span m="4606140">conclusion</span> <span m="4606780">that</span> <span m="4606900">probability</span> <span m="4607410">R</span> <span m="4607710">equals</span> <span m="4608040">K</span> <span m="4610290">is</span> <span m="4610750">this.</span> <span m="4612330">1</span> <span m="4612610">over</span> <span m="4612830">K</span> <span m="4613130">factorial</span> <span m="4614820">times</span> <span m="4615100">n</span> <span m="4615340">minus</span> <span m="4615930">K</span> <span m="4616300">down</span> <span m="4616530">here</span> <span m="4617490">for</span> <span m="4617630">K</span> <span m="4618190">less</span> <span m="4618590">than</span> <span m="4619800">or</span> <span m="4619970">equal</span> <span m="4620310">to</span> <span m="4620410">n</span> <span m="4620600">minus</span> <span m="4620920">2.</span> <span m="4622680">And</span> <span m="4622950">1</span> <span m="4623170">over</span> <span m="4623400">n</span> <span m="4623520">factorial</span> <span m="4625230">if</span> <span m="4625470">K</span> <span m="4625880">equals</span> <span m="4626270">n</span> <span m="4626430">minus</span> <span m="4626780">1</span> <span m="4627140">or</span> <span m="4627370">n.</span> </p>
<p><span m="4630130">Then</span> <span m="4630380">you</span> <span m="4630490">would</span> <span m="4630630">plug</span> <span m="4631060">that</span> <span m="4631410">nasty</span> <span m="4631820">looking</span> <span m="4632100">thing</span> <span m="4633530">into</span> <span m="4633820">there.</span> <span m="4634590">So</span> <span m="4634780">multiply</span> <span m="4635120">by</span> <span m="4635270">K.</span> <span m="4636110">And</span> <span m="4636270">you'd</span> <span m="4636340">have</span> <span m="4636480">to</span> <span m="4636560">sum it</span> <span m="4636900">up.</span> <span m="4638330">And</span> <span m="4638660">Lord</span> <span m="4638930">help</span> <span m="4639150">you.</span> <span m="4639780">That's</span> <span m="4640110">just</span> <span m="4640290">a</span> <span m="4640330">nightmare</span> <span m="4640700">to</span> <span m="4640780">do.</span> <span m="4642000">You'd</span> <span m="4642120">have</span> <span m="4642230">a</span> <span m="4642280">very</span> <span m="4642590">hard</span> <span m="4642790">time</span> <span m="4643060">getting</span> <span m="4643250">the</span> <span m="4643410">answer.</span> <span m="4644640">If</span> <span m="4644940">you</span> <span m="4645030">doubt</span> <span m="4645330">that,</span> <span m="4646110">try</span> <span m="4646650">it.</span> <span m="4647980">But</span> <span m="4648110">that</span> <span m="4648230">would</span> <span m="4648320">be</span> <span m="4648640">a</span> <span m="4648810">natural</span> <span m="4649270">way</span> <span m="4649410">to</span> <span m="4649500">proceed.</span> </p>
<p><span m="4652790">But</span> <span m="4652900">it</span> <span m="4652960">turns</span> <span m="4653300">out</span> <span m="4653490">there</span> <span m="4653620">is</span> <span m="4653730">a</span> <span m="4653820">trivial</span> <span m="4654370">way</span> <span m="4654520">to</span> <span m="4654580">get</span> <span m="4654730">the</span> <span m="4654870">answer.</span> <span m="4655430">And</span> <span m="4655530">this</span> <span m="4655660">is</span> <span m="4655770">a</span> <span m="4655840">very</span> <span m="4656350">powerful</span> <span m="4656810">technique</span> <span m="4657940">using</span> <span m="4658310">linearity</span> <span m="4658720">of</span> <span m="4658810">expectation.</span> <span m="4660200">And</span> <span m="4660380">for</span> <span m="4660530">sure</span> <span m="4660990">there</span> <span m="4661110">will</span> <span m="4661230">be</span> <span m="4661420">a</span> <span m="4661490">problem</span> <span m="4661850">on</span> <span m="4661950">the</span> <span m="4662020">final</span> <span m="4662350">exam</span> <span m="4663240">just</span> <span m="4663590">like</span> <span m="4663840">this.</span> <span m="4665790">And</span> <span m="4665910">so</span> <span m="4666050">if</span> <span m="4666140">you</span> <span m="4666240">go</span> <span m="4666450">down</span> <span m="4666690">this</span> <span m="4666860">path,</span> <span m="4667400">which</span> <span m="4667460">is</span> <span m="4667560">a</span> <span m="4667660">natural</span> <span m="4668150">first</span> <span m="4668440">path,</span> <span m="4669620">it</span> <span m="4669790">may</span> <span m="4669950">take</span> <span m="4670270">you</span> <span m="4671400">the</span> <span m="4671490">rest</span> <span m="4671710">of</span> <span m="4671770">the</span> <span m="4671830">day</span> <span m="4672880">to</span> <span m="4673020">solve</span> <span m="4673420">it.</span> <span m="4673950">But</span> <span m="4674120">the</span> <span m="4674190">method</span> <span m="4674470">I'm</span> <span m="4674530">going</span> <span m="4674650">to</span> <span m="4674710">show</span> <span m="4675000">you</span> <span m="4675130">will</span> <span m="4675250">take</span> <span m="4675500">you</span> <span m="4675760">a</span> <span m="4675820">couple</span> <span m="4676080">minutes</span> <span m="4676690">to</span> <span m="4676810">do</span> <span m="4677000">it.</span> </p>
<p><span m="4682370">Now,</span> <span m="4682810">the</span> <span m="4682890">trick</span> <span m="4683180">is</span> <span m="4683290">to</span> <span m="4683370">use</span> <span m="4683580">linearity</span> <span m="4684050">of</span> <span m="4684140">expectations.</span> <span m="4685070">So</span> <span m="4686160">the</span> <span m="4686300">problem</span> <span m="4686620">is</span> <span m="4686740">there's</span> <span m="4686920">no</span> <span m="4687110">sum</span> <span m="4687540">here.</span> <span m="4689610">So</span> <span m="4689810">what we</span> <span m="4689940">need</span> <span m="4690090">to</span> <span m="4690150">do</span> <span m="4690450">is</span> <span m="4690550">we're</span> <span m="4690750">going</span> <span m="4690880">to</span> <span m="4690940">express</span> <span m="4691470">R</span> <span m="4691960">as</span> <span m="4692290">the</span> <span m="4692380">sum</span> <span m="4693520">of</span> <span m="4693730">random</span> <span m="4694090">variables.</span> <span m="4696300">And</span> <span m="4696550">the</span> <span m="4696620">way</span> <span m="4696860">we're</span> <span m="4696970">going</span> <span m="4697090">to</span> <span m="4697150">do</span> <span m="4697390">that</span> <span m="4700940">is</span> <span m="4701030">as</span> <span m="4701170">follows.</span> <span m="4702740">And</span> <span m="4702850">it's</span> <span m="4702950">not</span> <span m="4703260">obvious,</span> <span m="4703750">but</span> <span m="4703870">once</span> <span m="4704050">you</span> <span m="4704130">see</span> <span m="4704400">it,</span> <span m="4704540">it's</span> <span m="4704740">easy</span> <span m="4705000">to</span> <span m="4705070">keep</span> <span m="4705330">using</span> <span m="4705660">it.</span> </p>
<p><span m="4708700">We</span> <span m="4708840">let</span> <span m="4709120">R</span> <span m="4709470">be</span> <span m="4709650">the</span> <span m="4709730">sum</span> <span m="4710080">of</span> <span m="4710230">R1</span> <span m="4710880">plus</span> <span m="4711260">R2</span> <span m="4713380">plus</span> <span m="4713670">Rn.</span> <span m="4715825">And</span> <span m="4716256">Ri</span> <span m="4718550">is</span> <span m="4718730">going</span> <span m="4718850">to</span> <span m="4718920">tell</span> <span m="4719310">us</span> <span m="4719550">the</span> <span m="4719710">event.</span> <span m="4720075">This is</span> <span m="4720440">sort</span> <span m="4720810">of</span> <span m="4720880">what</span> <span m="4721030">you</span> <span m="4721130">were</span> <span m="4722250">talking</span> <span m="4722570">about</span> <span m="4722750">before</span> <span m="4723110">with</span> <span m="4723230">the</span> <span m="4723300">event</span> <span m="4723590">of</span> <span m="4723660">a</span> <span m="4723730">boy</span> <span m="4724100">or</span> <span m="4724170">event</span> <span m="4724410">of</span> <span m="4724490">a</span> <span m="4724560">girl.</span> <span m="4725170">This</span> <span m="4725380">is</span> <span m="4725470">going</span> <span m="4725590">to</span> <span m="4725670">be</span> <span m="4725810">the</span> <span m="4725960">event</span> <span m="4726550">of</span> <span m="4726640">the</span> <span m="4726800">ith</span> <span m="4727190">man</span> <span m="4727580">gets</span> <span m="4727800">his</span> <span m="4727980">right</span> <span m="4728230">hat</span> <span m="4728480">back.</span> <span m="4730830">So</span> <span m="4731000">it's</span> <span m="4731100">an</span> <span m="4731180">indicator</span> <span m="4731770">random</span> <span m="4732070">variable.</span> <span m="4732710">It's</span> <span m="4732900">1</span> <span m="4734740">if</span> <span m="4734990">the</span> <span m="4735120">ith</span> <span m="4735480">man</span> <span m="4738630">gets</span> <span m="4738920">the</span> <span m="4739020">right</span> <span m="4739260">hat.</span> <span m="4742170">And</span> <span m="4742480">it's 0</span> <span m="4743620">if</span> <span m="4743760">he</span> <span m="4743840">doesn't.</span> </p>
<p><span m="4750060">So</span> <span m="4750640">whenever</span> <span m="4751960">the</span> <span m="4752190">ith guy</span> <span m="4752480">gets</span> <span m="4752670">his</span> <span m="4752790">hat</span> <span m="4753010">back,</span> <span m="4753210">that</span> <span m="4753340">counts</span> <span m="4753610">as</span> <span m="4753730">one,</span> <span m="4754700">and</span> <span m="4754850">now</span> <span m="4754980">you</span> <span m="4755100">can</span> <span m="4755240">see</span> <span m="4755510">why</span> <span m="4755690">this</span> <span m="4755850">sum</span> <span m="4756190">works.</span> <span m="4757095">R</span> <span m="4757820">is</span> <span m="4757940">the</span> <span m="4758040">number</span> <span m="4759020">of</span> <span m="4759180">men</span> <span m="4759440">to</span> <span m="4759500">get</span> <span m="4759640">the</span> <span m="4759720">right</span> <span m="4759940">hat</span> <span m="4760190">back.</span> <span m="4761110">And</span> <span m="4761270">it</span> <span m="4761370">basically</span> <span m="4761980">there's</span> <span m="4762150">a</span> <span m="4762210">one</span> <span m="4762620">counted</span> <span m="4763080">in</span> <span m="4763210">here</span> <span m="4763540">every</span> <span m="4763840">time</span> <span m="4764210">a</span> <span m="4764260">guy</span> <span m="4764470">gets</span> <span m="4764680">his</span> <span m="4764800">hat</span> <span m="4765020">back,</span> <span m="4766210">and</span> <span m="4766400">0</span> <span m="4766900">if</span> <span m="4766980">he</span> <span m="4767070">doesn't.</span> <span m="4768740">So</span> <span m="4768960">this</span> <span m="4769170">sum</span> <span m="4769840">is</span> <span m="4770030">counting</span> <span m="4771250">how</span> <span m="4771540">many</span> <span m="4771740">men</span> <span m="4772000">got</span> <span m="4772180">their</span> <span m="4772430">right</span> <span m="4772630">hat</span> <span m="4772840">back.</span> <span m="4776280">Non-obvious</span> <span m="4777070">the</span> <span m="4777140">first</span> <span m="4777430">time,</span> <span m="4778070">gets</span> <span m="4778280">really</span> <span m="4778540">simple</span> <span m="4778980">the</span> <span m="4779210">fourth</span> <span m="4779480">or</span> <span m="4779530">fifth</span> <span m="4779770">time.</span> <span m="4781420">We'll</span> <span m="4781540">try</span> <span m="4781710">to</span> <span m="4781810">do</span> <span m="4781950">a</span> <span m="4782000">couple</span> <span m="4782310">of</span> <span m="4782380">them</span> <span m="4783430">today.</span> </p>
<p><span m="4788290">All</span> <span m="4788490">right,</span> <span m="4788740">now</span> <span m="4788980">the</span> <span m="4789090">expected</span> <span m="4789730">value</span> <span m="4790430">of</span> <span m="4790570">R</span> <span m="4791860">is</span> <span m="4792050">easy.</span> <span m="4795230">It's</span> <span m="4795420">just</span> <span m="4795730">by</span> <span m="4795850">linearity of</span> <span m="4796290">expectation,</span> <span m="4797270">expected</span> <span m="4797670">value of</span> <span m="4797940">R1</span> <span m="4799590">and</span> <span m="4799720">so</span> <span m="4799960">forth</span> <span m="4800590">plus</span> <span m="4800840">the</span> <span m="4800900">expected</span> <span m="4801340">value</span> <span m="4801590">of</span> <span m="4801700">Rn.</span> <span m="4805310">The</span> <span m="4805490">expected</span> <span m="4806140">value</span> <span m="4806780">of</span> <span m="4806950">an</span> <span m="4807160">indicator</span> <span m="4808120">random</span> <span m="4808470">variable</span> <span m="4810150">is</span> <span m="4810330">just</span> <span m="4810570">the</span> <span m="4810640">probability</span> <span m="4811240">that</span> <span m="4811350">it's</span> <span m="4811490">1.</span> <span m="4813350">Right</span> <span m="4813590">it's</span> <span m="4813920">1</span> <span m="4814380">times</span> <span m="4814620">the</span> <span m="4814680">probability,</span> <span m="4815040">it's</span> <span m="4815150">0</span> <span m="4816030">times</span> <span m="4816290">the</span> <span m="4816340">probability</span> <span m="4816700">of</span> <span m="4816790">0.</span> <span m="4817520">That's</span> <span m="4817780">just</span> <span m="4817970">the</span> <span m="4818040">probability</span> <span m="4818520">that</span> <span m="4818610">it's</span> <span m="4818720">1.</span> </p>
<p><span m="4828650">What's</span> <span m="4828990">the</span> <span m="4829090">probability</span> <span m="4829760">that</span> <span m="4830260">the</span> <span m="4830400">first</span> <span m="4830720">man</span> <span m="4831000">gets</span> <span m="4831210">the</span> <span m="4831290">right</span> <span m="4831480">hat</span> <span m="4831680">back?</span> <span m="4834400">1</span> <span m="4834640">over</span> <span m="4834890">n.</span> <span m="4835080">There's</span> <span m="4835230">n</span> <span m="4835390">hats.</span> <span m="4835730">He</span> <span m="4835790">gets</span> <span m="4835980">a</span> <span m="4836050">random</span> <span m="4836390">one.</span> <span m="4838820">What's</span> <span m="4839140">the</span> <span m="4839230">probability</span> <span m="4839680">that</span> <span m="4839780">the</span> <span m="4839880">second</span> <span m="4840190">man</span> <span m="4840480">gets</span> <span m="4840690">his</span> <span m="4840890">hat</span> <span m="4841120">back?</span> <span m="4844870">Not</span> <span m="4845080">1</span> <span m="4845270">over</span> <span m="4845480">n</span> <span m="4845610">minus</span> <span m="4846050">1.</span> <span m="4847980">He's</span> <span m="4848720">coming</span> <span m="4849050">in</span> <span m="4849300">whether</span> <span m="4849470">he's</span> <span m="4849600">first,</span> <span m="4849910">second,</span> <span m="4850190">or</span> <span m="4850290">last.</span> <span m="4850760">He</span> <span m="4850840">gets</span> <span m="4851040">a</span> <span m="4851140">random</span> <span m="4851690">hat.</span> <span m="4853630">1</span> <span m="4853990">in</span> <span m="4854070">n</span> <span m="4854250">chance</span> <span m="4854650">it's</span> <span m="4854820">his.</span> </p>
<p><span m="4859340">What's</span> <span m="4859590">the</span> <span m="4859700">probability</span> <span m="4860200">the</span> <span m="4860280">last</span> <span m="4860870">man</span> <span m="4861250">gets</span> <span m="4861460">the</span> <span m="4861560">right</span> <span m="4861790">hat</span> <span m="4862050">back?</span> <span m="4864660">One</span> <span m="4864890">over n.</span> <span m="4865200">Doesn't</span> <span m="4865440">really</span> <span m="4865660">matter</span> <span m="4865900">if he's</span> <span m="4866030">first</span> <span m="4866320">or</span> <span m="4866380">last.</span> <span m="4866790">Just</span> <span m="4867000">sort</span> <span m="4867150">of</span> <span m="4867360">tricks</span> <span m="4867790">up</span> <span m="4867980">a</span> <span m="4868020">little</span> <span m="4868220">bit.</span> <span m="4869030">He's</span> <span m="4869250">getting</span> <span m="4869560">a</span> <span m="4869640">random</span> <span m="4870000">hat</span> <span m="4870250">back.</span> <span m="4871480">So</span> <span m="4871640">it's</span> <span m="4871730">1</span> <span m="4871920">over</span> <span m="4872030">n.</span> <span m="4875330">I</span> <span m="4875450">got</span> <span m="4875710">n</span> <span m="4875890">of</span> <span m="4875970">each,</span> <span m="4876770">1</span> <span m="4876940">over</span> <span m="4877205">n,</span> <span m="4877470">so</span> <span m="4877610">what's</span> <span m="4877880">the</span> <span m="4877980">expected</span> <span m="4878480">number</span> <span m="4878750">of</span> <span m="4878830">men</span> <span m="4879040">to</span> <span m="4879100">get</span> <span m="4879240">the</span> <span m="4879320">right</span> <span m="4879480">hat</span> <span m="4879700">back?</span> <span m="4881170">One.</span> <span m="4883290">Now,</span> <span m="4883430">the</span> <span m="4883520">math</span> <span m="4883820">doesn't</span> <span m="4884190">get</span> <span m="4884380">much</span> <span m="4884590">easier</span> <span m="4884950">than</span> <span m="4885080">that.</span> </p>
<p><span m="4886680">Now,</span> <span m="4886870">the</span> <span m="4886990">amazing</span> <span m="4887570">thing</span> <span m="4888910">is</span> <span m="4889100">we</span> <span m="4889240">just</span> <span m="4889480">proved</span> <span m="4891320">that</span> <span m="4891450">if</span> <span m="4891520">we</span> <span m="4891620">take</span> <span m="4891870">that</span> <span m="4892090">mess,</span> <span m="4893110">stick</span> <span m="4893460">it</span> <span m="4893540">in</span> <span m="4893650">here</span> <span m="4894730">and</span> <span m="4894920">sum</span> <span m="4895200">it</span> <span m="4895280">up,</span> <span m="4896460">what</span> <span m="4896650">answer</span> <span m="4896910">do</span> <span m="4896990">we</span> <span m="4897080">get?</span> <span m="4898610">1.</span> <span m="4899900">That</span> <span m="4900130">is</span> <span m="4900220">certainly</span> <span m="4900550">not</span> <span m="4900790">obvious,</span> <span m="4901200">but</span> <span m="4901330">that</span> <span m="4901500">is</span> <span m="4901590">a</span> <span m="4901640">consequence</span> <span m="4902310">of</span> <span m="4902400">everything</span> <span m="4902700">we've</span> <span m="4902820">just</span> <span m="4903050">done.</span> <span m="4904240">We've</span> <span m="4904370">just</span> <span m="4904540">given</span> <span m="4904700">a</span> <span m="4904760">probability</span> <span m="4905390">proof</span> <span m="4906230">of</span> <span m="4906360">that</span> <span m="4906570">fact.</span> </p>
<p><span m="4907560">But</span> <span m="4907730">the</span> <span m="4907800">nice</span> <span m="4908040">thing</span> <span m="4908190">here</span> <span m="4908440">is</span> <span m="4909780">there's</span> <span m="4909930">actually</span> <span m="4910170">even</span> <span m="4910350">more</span> <span m="4910510">powerful.</span> <span m="4912920">Did</span> <span m="4913070">I</span> <span m="4913230">need</span> <span m="4913420">to</span> <span m="4913520">assume</span> <span m="4914180">that</span> <span m="4914510">it</span> <span m="4914660">was</span> <span m="4914770">a</span> <span m="4914860">random</span> <span m="4915270">permutation</span> <span m="4915990">of</span> <span m="4916080">hats</span> <span m="4916870">like</span> <span m="4917090">I</span> <span m="4917130">would</span> <span m="4917490">need</span> <span m="4917740">to</span> <span m="4917810">assume</span> <span m="4918110">for</span> <span m="4918220">this?</span> <span m="4923180">No</span> <span m="4923450">independence</span> <span m="4924040">is</span> <span m="4924150">needed.</span> <span m="4926040">In</span> <span m="4926200">fact,</span> <span m="4926620">there's</span> <span m="4926800">all</span> <span m="4927050">sorts</span> <span m="4927540">of</span> <span m="4927630">distributions</span> <span m="4929110">that</span> <span m="4929250">will</span> <span m="4929350">give</span> <span m="4929570">the</span> <span m="4929640">same</span> <span m="4929990">expected</span> <span m="4930450">value.</span> <span m="4931140">All</span> <span m="4931680">I</span> <span m="4931880">need</span> <span m="4932290">is</span> <span m="4932400">that</span> <span m="4932510">each</span> <span m="4932820">person</span> <span m="4933170">gets</span> <span m="4933350">the</span> <span m="4933450">right</span> <span m="4933680">hat</span> <span m="4933980">back</span> <span m="4934220">with</span> <span m="4934340">probability</span> <span m="4935430">1</span> <span m="4935630">over n.</span> <span m="4937246">In</span> <span m="4937650">fact,</span> <span m="4937840">this</span> <span m="4937990">is</span> <span m="4938110">an</span> <span m="4938180">example</span> <span m="4938670">of</span> <span m="4938740">a</span> <span m="4938790">different</span> <span m="4939260">distribution</span> <span m="4939960">for</span> <span m="4940080">which</span> <span m="4940340">the</span> <span m="4940450">result</span> <span m="4940830">is</span> <span m="4940950">the</span> <span m="4941040">same.</span> </p>
<p><span m="4942390">Say</span> <span m="4942680">you're</span> <span m="4942820">at</span> <span m="4942890">a</span> <span m="4942920">Chinese</span> <span m="4943400">restaurant.</span> <span m="4944680">And you know</span> <span m="4944860">they</span> <span m="4945150">have</span> <span m="4945360">the</span> <span m="4945540">thing</span> <span m="4945740">that</span> <span m="4945850">spins</span> <span m="4946250">in</span> <span m="4946310">the</span> <span m="4946370">middle</span> <span m="4946540">of</span> <span m="4946600">the</span> <span m="4946660">table?</span> <span m="4948640">Say</span> <span m="4948890">that</span> <span m="4948980">you</span> <span m="4949090">each</span> <span m="4949420">order</span> <span m="4949650">an</span> <span m="4949730">appetizer.</span> <span m="4950380">There's</span> <span m="4950560">n</span> <span m="4950750">people</span> <span m="4951060">and they're</span> <span m="4951130">around</span> <span m="4951480">a big</span> <span m="4951760">circular</span> <span m="4952150">table.</span> <span m="4952850">Everybody</span> <span m="4953280">gets</span> <span m="4953490">their</span> <span m="4953860">appetizer.</span> <span m="4955110">Wonton</span> <span m="4955530">soup,</span> <span m="4956010">whatever.</span> </p>
<p><span m="4956930">And</span> <span m="4957020">then</span> <span m="4957110">there's</span> <span m="4957270">always</span> <span m="4957570">the</span> <span m="4957650">joker</span> <span m="4958110">who</span> <span m="4958240">spins</span> <span m="4959310">the</span> <span m="4959420">thing</span> <span m="4960540">and</span> <span m="4960770">spins</span> <span m="4961150">around</span> <span m="4962570">and</span> <span m="4962720">then</span> <span m="4962830">it</span> <span m="4962930">stops</span> <span m="4964560">and</span> <span m="4964720">now</span> <span m="4964820">you've</span> <span m="4964950">got</span> <span m="4965050">a</span> <span m="4965140">random</span> <span m="4965580">appetizer</span> <span m="4966140">in</span> <span m="4966220">front</span> <span m="4966410">of</span> <span m="4966490">you.</span> <span m="4968480">In</span> <span m="4968770">this</span> <span m="4968950">case,</span> <span m="4969150">we</span> <span m="4969230">want</span> <span m="4969400">to</span> <span m="4969460">know</span> <span m="4970690">what's</span> <span m="4970930">the</span> <span m="4971010">expected</span> <span m="4971490">number</span> <span m="4971740">of</span> <span m="4971820">people</span> <span m="4972180">to</span> <span m="4972270">get</span> <span m="4972440">the</span> <span m="4972530">right</span> <span m="4972960">appetizer</span> <span m="4973570">back.</span> <span m="4973920">Not</span> <span m="4974680">the</span> <span m="4974800">other</span> <span m="4974940">guys</span> <span m="4975180">wonton</span> <span m="4975600">soup,</span> <span m="4975910">but</span> <span m="4976010">yours.</span> <span m="4978460">That's</span> <span m="4978700">a</span> <span m="4978740">different</span> <span m="4979210">probability</span> <span m="4979740">space</span> <span m="4980120">because</span> <span m="4980270">there's</span> <span m="4980420">only</span> <span m="4980710">n</span> <span m="4981000">sample</span> <span m="4981400">points,</span> <span m="4982180">n</span> <span m="4982430">places</span> <span m="4982800">where the</span> <span m="4982970">thing could</span> <span m="4983190">have</span> <span m="4983260">stopped.</span> <span m="4983570">Not</span> <span m="4983780">n</span> <span m="4983900">factorial</span> <span m="4985220">like</span> <span m="4985420">the</span> <span m="4985500">hats.</span> </p>
<p><span m="4988390">Well,</span> <span m="4989270">does</span> <span m="4989440">the</span> <span m="4989520">analysis</span> <span m="4990040">change?</span> <span m="4992670">Exactly</span> <span m="4993430">the</span> <span m="4993530">same.</span> <span m="4994660">Ri</span> <span m="4995410">is</span> <span m="4995550">the</span> <span m="4995650">indicator</span> <span m="4996080">variable</span> <span m="4996460">for</span> <span m="4996570">the</span> <span m="4996670">ith</span> <span m="4996960">person</span> <span m="4997320">gets</span> <span m="4997520">the</span> <span m="4997620">right</span> <span m="4998070">appetizer</span> <span m="4998700">back.</span> <span m="5000200">Linearity</span> <span m="5000640">of</span> <span m="5000730">expectation.</span> <span m="5002650">The</span> <span m="5002780">expected</span> <span m="5003240">value</span> <span m="5003560">of</span> <span m="5003620">the</span> <span m="5003680">indicator</span> <span m="5004130">variable</span> <span m="5004580">is</span> <span m="5004690">just</span> <span m="5004930">the</span> <span m="5005010">probability</span> <span m="5005500">that</span> <span m="5005610">it's</span> <span m="5005730">1.</span> <span m="5006750">And</span> <span m="5006870">the</span> <span m="5006930">probability</span> <span m="5007470">that</span> <span m="5007630">any</span> <span m="5007830">person</span> <span m="5008210">gets</span> <span m="5008390">the</span> <span m="5008490">right</span> <span m="5008690">appetizer is</span> <span m="5009320">1</span> <span m="5009500">over</span> <span m="5009710">n.</span> </p>
<p><span m="5011010">So</span> <span m="5011160">the</span> <span m="5011280">answer</span> <span m="5011620">is</span> <span m="5011700">the</span> <span m="5011780">same.</span> <span m="5012150">The</span> <span m="5012210">expected</span> <span m="5012710">number</span> <span m="5012910">of</span> <span m="5012970">people</span> <span m="5013220">to</span> <span m="5013290">get</span> <span m="5013400">the</span> <span m="5013490">right</span> <span m="5013690">appetizer</span> <span m="5014210">back</span> <span m="5014630">is</span> <span m="5014800">1.</span> <span m="5016400">So</span> <span m="5016620">totally</span> <span m="5017130">different</span> <span m="5017880">probability</span> <span m="5018430">spaces,</span> <span m="5019440">exactly</span> <span m="5020130">the</span> <span m="5020220">same</span> <span m="5020490">analysis</span> <span m="5021980">and</span> <span m="5022210">answer.</span> <span m="5023840">OK,</span> <span m="5024150">so</span> <span m="5024250">we'll</span> <span m="5024350">do</span> <span m="5024490">more</span> <span m="5024740">of</span> <span m="5024810">this</span> <span m="5025410">next</span> <span m="5025710">time.</span> </p>
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